Hierarchy spectrum of SM fermions: from top quark to electron neutrino
Received: October
Hierarchy spectrum of SM fermions: from top quark to electron neutrino
SheSheng 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 fourfermion operators of EinsteinCartan type The study is motivated by the speculation that these fourfermion operators are probably originated due to the quantum gravity, which provides the natural regularization for chiralsymmetric gauge eld theories. In the chiralgauge symmetry breaking phase, as to achieve the energetically favorable ground state, only the topquark mass is generated via the spontaneous symmetry breaking, and other fermion masses are generated via the explicit symmetry breaking induced by the topquark mass, fourfermion interactions and fermion avor mixing matrices. A phase transition from the symmetry breaking phase to the chiralgauge 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 righthanded neutrinos Rf breaking are originated by the fourfermion 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 forwardbackward asymmetry of ttproduction, as well as remarks on the candidates of light and heavy dark matter particles ArXiv ePrint: 1605.01266
to; electron; neutrino; netuning; (fermions; scalar and pseudoscalar bosons)

nicolor and Composite Models
1 Introduction
Fourfermion operators beyond the SM
Regularization and quantum gravity
EinsteinCartan theory with the SM gauge symmetries and fermion content
SM gaugesymmetric fourfermion operators
Fourfermion operators of quarklepton interactions
Gauge vs mass eigenstates in fermionfamily space
Quarklepton interaction sector
Spontaneous symmetry breaking
xedpoint domain and only topquark mass generated via the SSB
The htticondensate model
The scaling region of the IRstable xed point
Experimental indications of composite Higgs boson?
Origins of explicit symmetry breaking
Quarklepton interactions
W boson coupling to righthanded fermions
SchwingerDyson equations for fermion selfenergy functions
Chiral symmetrybreaking 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 massgap equations for the third family
Fermion masses and running Yukawa couplings
Approximate fermion massgap equations of the second family
Running fermion masses and Yukawa couplings
Approximate massgap 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 darkmatter particles
Introduction
The parityviolating (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
Higgsboson model or the NambuJonaLasinio (NJL) [1] with e ective fourfermion
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 Higgsboson mass 126 GeV [8, 9] and topquark
mass 173 GeV [10, 11], as well as the other SM fermion masses and familymixing 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 familymixing matrices are equally
fundamental, and closely related. Since Gatto et al. [12] tried to
nd the relation between
the Cabibbo mixing angle and lightquark masses, the tremendous e ort and many models
have been made to study the relation of the SM fermion masses and familymixing 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 fermionmass matrices in quark
and/or lepton sectors to nd the fermionfamily mixing matrices as functions of observed
fermion masses, i.e., the eigenvalues of the original fermionmass matrices. Whereas some
other models try to nd the relations of fermion masses and familymixing matrices on the
basis of theoretically modelbuilding approaches, for example, the leftright 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 modelindependent approach, the
fermionmass 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 nontrivial 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 longstanding problem by considering e ective
fourfermion operators in the framework of the SM gauge symmetries and fermion
content: neutrinos, charged leptons and quarks. In order to accommodate highdimensional
operators of fermion elds in the SMframework of a wellde ned quantum
eld theory at
the highenergy 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
IRdomain and UVdomain (scaling regions) of these stable IR and UV
xed points, which
operators become physically relevant (e ectively dimension4) 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
fourfermion operator
L = Lkinetic + G( LiatRa)(tbR Lib);
of Bardeen, Hill and Lindner (BHL) htticondensate model [81] in the context of a
wellde ned quantum
eld theory at the highenergy scale . The fourfermion operator (1.1)
undergoes the spontaneous symmetry breaking (SSB) dynamics responsible for the
generation of topquark and Higgsboson masses in the domain of IRstable 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 lowenergy SM physics,
including the values of topquark and Higgsboson masses, was supposed to be achieved by the
RGequations in the domain of the IRstable 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
lowenergies. 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 highenergy scale
, since then, many models based on this idea have been studied [86, 87].
Nowadays, the known topquark and Higgs boson masses completely determine the
boundary conditions of the RG equations for the topquark Yukawa coupling gt( ) and
Higgsboson quartic coupling ~( ) in the composite Higgs boson model (1.1). Using the
experimental values of topquark and Higgs boson masses, we obtained [88, 89] the unique
solutions gt( ) and ~( )to these RG equations, provided the appropriate nonvanishing
where the e ective quartic coupling ~( ) of composite Higgs bosons vanishes.
The formfactor 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 fourfermion 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
secondorder 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
netuning (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 quantumgravity origin
of fourfermion operators at the cuto
, the BHL htticondensate model and the SSB,
we show that due to fourfermion operators (i) there are the SM gauge symmetric
vertexes of quarklepton interactions; (ii) the oneparticleirreducible (1PI) vertexfunction
boson coupling becomes approximately vectorlike at TeV scale. Both interacting
vertexes contribute the explicit symmetry breaking (ESB) terms to the SchwingerDyson
(SD) equations of fermion selfenergy functions. As a result, once the topquark mass is
generated via the SSB, the masses of third fermion family ( ; ; b) are generated by the
ESB via quarklepton interactions and W boson vectorlike coupling. Within the third
fermion family, we qualitatively study the hierarchy of fermion masses and e ective Yukawa
couplings in terms of the topquark 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 vertexfunctions of quarklepton interactions and approximately
vectorlike W boson coupling among three fermion families at TeV scale. As a consequence,
these 1PI vertexfunctions introduce the ESB terms into the SDequations of the fermion
selfenergy functions for all SM fermions in three families. Once the topquark mass is
generated via the SSB, all other SM fermions acquire their masses via the ESB terms by
(i) fourfermion 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 familymixing matrices, and
all fermion masses and Yukawa couplings are functions of the topquark 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 highenergy 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 fourfermion
operators, including quarklepton interactions. In section 3, we describe fermion avor
mixing matrices in lepton and quark sectors, as well as quarklepton interaction sector. In
section 4, we give a brief recall that the SSB is responsible only for the topquark 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
Fourfermion 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 leptonquark 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 xedpoint domain of weak fourfermion
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 chiralgauge
interactions should be wellde ned by including the quantum gravity that naturally
provides a spacetime regularization (UV cuto ). As an example, the nite superstring theory
is proposed by postulating that instead of a simple spacetime point, the fundamental
spacetime \constituents" is a spacetime \string". The Planck scale is a plausible cuto ,
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 highdimensional operators. Instead, as a postulation or motivation, we argue the
presence of at least fourfermion operators beyond the SM from the following point view.
A wellde ned quantum
eld theory for the SM Lagrangian requires a natural
regularization (UV cuto
) fully preserving the SM chiralgauge symmetry.
The quantum
gravity naturally provides a such regularization of discrete spacetime 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 nogo theorem [101{104] 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 spacetime cuto
a~. This implies that the natural quantumgravity regularization for
the SM should lead us to consider at least dimension6 fourfermion 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 fourfermion
operators of the torsionfree EinsteinCartan 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.
EinsteinCartan theory with the SM gauge symmetries and fermion
conThe Lagrangian of torsionfree EinsteinCartan (EC) theory reads,
LEC(e; !; ) = LEC(e; !) +
= @
ig! and the axial
current J d =
of massless fermion
elds. The fourfermion coupling G relates to
the gravitationfermion gauge coupling g and fundamental spacetime cuto a~.
Within the SM fermion content, we consider massless left and righthanded Weyl
Rf carrying quantum numbers of the SM symmetries, as well as three
righthanded Weyl sterile neutrinos
Rf and their lefthanded conjugated
f c =
i 2( R ) , where \f " is the fermionfamily index. Analogously to the EC theory (2.1),
we obtain a torsionfree, di eomorphism and local gaugeinvariant 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 fourfermion coupling G is unique for all fourfermion operators and highdimensional
fermion operators (d > 6) are neglected.
eq. (2.2) can be written as [107]
By using the Fierz theorem [105, 106], the dimension6 fourfermion 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 spacetime cuto , in addition to
dimension6 fourfermion operators, there are highdimensional 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 dimension4. Thus
the torsionfree EC theory with the attractive fourfermion 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 fourcomponent Dirac fermions
Weyl neutrinos Lf and fourcomponent sterile Majorana neutrinos Mf = ( Rfc + Rf
f = ( Lf +
Rf), twocomponent righthanded
kinetic terms read
Eq. (2.6) preserves not only the SM gauge symmetries and global fermionfamily
symmetries, but also the global symmetries for fermionnumbers conservations. We adopt the
e ective fourfermion operators (2.6) in the context of a wellde ned quantum eld theory
at the highenergy scale .
SM gaugesymmetric fourfermion operators
explicitly show SM gauge symmetric fourfermion operators in eq. (2.6). In the quark
sector, the fourfermion 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 fourfermion
operators of topquark channel [81] and bottomquark 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].
fourfermion 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 leptonnumber 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 fourfermion 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 ;
Fourfermion operators of quarklepton interactions
Although the fourfermion operators in eq. (2.6) do not have quarklepton interactions,
we consider the following SM gaugesymmetric fourfermion 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 fourfermion operators (2.11) of quarklepton 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 EinsteinCartan 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 quarklepton interacting operators (2.11) to
G X n(`iLf efR0 )(daRf0 Lfia) + (`iLf efR0)(uaRf0 Lfia) ;
o
analogously to the fourfermion operators in eq. (2.6).
Gauge vs mass eigenstates in fermionfamily space
Due to the unique fourfermion coupling G and the global fermionfamily UL(3)
symmetry of eq. (2.6), one is allowed to perform chiral transformations UL 2 UL(3) and
without the family avormixing 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 fermionfamily space the fourfermion operators (2.6) and twofermion operators (
the latter is relating to fermion mass matrices.
For the quark sector, the fourfermion 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 uquark sector uf0
R ) dfR0 represented by the dquark sector df0
f0
Due to the unique fourfermion coupling G and the global fermionfamily ULu(3)
URu(3) symmetry for the uquark sector and ULd(3)
URd (3) symmetry for the dquark
sector in eq. (3.1), we perform four unitary chiral transformations from gaugeeigenstates
to masseigenstates:
sector and the dquark sector. As a result, all quark elds are mass eigenstates, the
fourfermion operators (3.1) are \diagonal" only for each quark family without familymixing,
In this representation, the vacuum expectation values of twofermion operators h Rf Lf i +
h:c:, i.e., quarkmass matrices are diagonalized in the fermionfamily 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 quarkfamily 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 fourfermion 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 fourfermion operators (3.9) are
\diagonal" only for each lepton family without familymixing,
h`fL `fR `fR `fL + (`fL Rf)( Rf`fL) + ( Rfc Rf )( Rf Rfc)i ;
and the vacuum expectation values of twolepton operators h`fR `fL i + h:c:, h Lf Rfi + h:c: and
R i + h:c:, i.e., leptonmass matrices are diagonalized in the fermionfamily 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 masseigenstates 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
leptonfamily 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 CPviolation 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 CPconservation 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 eigenvectors. 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 fourfermion
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
Quarklepton 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 fermionfamily space the fourfermion 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 fermionfamily 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 fourfermion
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, twofermion mass operators and
fourfermion operators are \diagonal" in the fermionfamily space.
Spontaneous symmetry breaking
In this section, we brie y recall and discuss that in the IRdomain of the IRstable xed
point Gc, the relevant fourfermion operator (2.8) undergoes the SSB and becomes an e
ectively bilinear and renormalizable Lagrangian that follows the RGequations to approach
the SM physics in the lowenergy. This is necessary and fundamental for studying the
origin of SM fermion masses in this article.
xedpoint domain and only topquark mass generated via the
Apart from what is possible new physics at the scale
explaining the origin of these
e ective fourfermion 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
dimensional4 renormalizable operators following RG equations (scaling laws), while other
irrelevant operators are suppressed by the cuto at least O(
In the IRdomain of the IRstable
xed point Gc, the fourfermion operator (1.1)
was shown [81] to become physically relevant and renormalizable operators of e ective
dimension4, due to the SSB dynamics of NJLtype. Namely, the Lagrangian (1.1) becomes
the e ective SM Lagrangian with bilinear topquark mass term and Yukawacoupling to the
composite Higgs boson H, which obeys the RGequations approaching to the lowenergy
SM physics characterized by the energy scale v
239:5 GeV. In addition, the topquark
and composite Higgsboson masses are correctly obtained by solving RGequations with
the appropriate nonvanishing formfactor of the Higgs boson in TeV scales [88, 89].
It seems that via the SSB dynamics the fourfermion operator the quarkcondensation
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 groundstate energy, and thus make
the topquark 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 fourfermion 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 massgap
equations of NJL type. Actually, the rst and second fourfermion operators in eq. (2.9)
e ective fourquark 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 IRdomain where the SSB occurs, except the top quark, all quarks
and leptons are massless and their fourfermion operators (3.4) and (3.12), as well as
repulsive fourfermion operators (2.4), are irrelevant dimension6 operators. Their
treelevel amplitudes of fourfermion 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 thirdfamily quarks and leptons are grouped together for their heavy masses,
due to the fermions
; ; b have the largest mixing with the top quark.
The htticondensate model
We brie y recall the BHL htticondensate model [81] for the full e ective Lagrangian of
the lowenergy SM in the IRdomain, 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 IRstable
Using the approach of large Ncexpansion with a xed value GNc, it is shown [81] that
the topquark channel of operators (2.8) undergoes the SSB dynamics in the IRdomain of
IRstable xed point Gc, leading to the generation of topquark mass
mt =
(1=2Nc)G Xhtatai =
by the htticondensate. As a result, the 2divergence (tadpolediagram) is removed by the
mass gapequation, the topquark channel of fourfermion operator (1.1) becomes physically
relevant and renormalizable operators of e ective dimension4. Namely, the e ective SM
Lagrangian with the bilinear topquark mass term and Yukawa coupling to the composite
Higgs boson H at the lowenergy scale
is given by [81]
L = Lkinetic + gt0( LtRH + h:c:) +
all renormalized quantities received fermionloop contributions are de ned with respect to
the lowenergy scale . The conventional renormalization Z
= 1 for fundamental fermions
and the unconventional wavefunction 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 wavefunction renormalization Z~H ( ), the Higgs
In the IRdomain where the SM of particle physics is realized, the full oneloop 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 topquark and Higgsboson masses, mt
173 GeV and mH
126 GeV, i.e., the massshell 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 Higgsboson model [81], we nd [88, 89]
the e ective topquark 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
Higgsboson 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 netuning problem.
Experimental indications of composite Higgs boson?
To end this section, we discuss the experimental indications of composite Higgs
boson. In the IRdomain, the dynamical symmetry breaking of fourfermion operator
Lib) of the topquark channel (2.8) accounts for the masses of top quark,
W and Z bosons as well as a Higgs boson composed by a topquark pair (tt) [81]. It
is shown [88, 89] that this mechanism consistently gives rise to the topquark and Higgs
masses, provided the appropriate value of nonvanishing formfactor of composite Higgs
boson at the highenergy 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
lowenergy 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 nonvanishing
formfactor Z~H ( ) means that after conventional wavefunction and vertex
renormaliza0 [see eqs. (4.2) and (4.3)], the composite
Higgs boson behaves as an elementary particle. The nonvanishing formfactor of
compostic coupling ~( ) monotonically decrease with the energy scale
ite Higgs boson is in fact related to the e ective Yukawacoupling 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 crosssections of the following three dominate processes of Higgsboson production
and decay [8, 108] or other relevant processes. Twogluon fusion produces a Higgs boson
via a topquark loop, which is proportional to the e ective Yukawa coupling gt( ). Then,
the produced Higgs boson decays into the twophoton state by coupling to a topquark
loop, and into the fourlepton state by coupling to two massive W bosons or two massive
Zbosons. Due to the t tcomposite nature of Higgs boson, the oneparticleirreducible
(1PI) vertexes of Higgsboson coupling to a topquark loop, two massive W bosons or two
massive Zbosons are proportional to the e ective Yukawa coupling gt( ). As a result,
both the Higgsboson 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 Higgsboson energy, representing the
total energy of nal states, e.g., twophoton state and fourlepton states, into which the
produced Higgs boson decays.
These discussions imply that the resonant amplitude (number of events) of twophoton
invariant mass m
126 GeV and/or fourlepton invariant mass m4l
126 GeV is
exof nal twophoton and/or fourlepton states increases, when the CM energy p
pected to become smaller as the produced Higgsboson 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 Higgsboson energy
= mt
and the SM value of Yukawa coupling gt2(mt) = 2mt2=v
gure 3). In this
scenario of composite Higgs boson, as the Higgsboson 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 bottomquark, taulepton and other fermions also weakly
decrease with increasing Higgsboson energy, this implies a slight decrease of number of
dilepton events in the DrellYan 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 quarklepton interactions (2.12) and fermionfamily mixing. We henceforth indicate
the SSBgenerated topquark mass mtsb and ESBgenerated masses mefb of other fermions,
they represent bare masses at the cuto energy scale E of the symmetry breaking phase.
Quarklepton 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 , fourfermion operators (3.21) contribute, via the tadpole
lepton interactions (2.11) that contribute quark and lepton ESB masses meb to massgap equations
or SDequations (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 massgap equations in the lepton
sector. Vice versa once leptons acquire their bare masses, via the same tadpole diagram
in gure 2, fourfermion operators (3.21) contribute the bare mass terms meub and medb in
massgap 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 nondiagonal mass matrices are eqs. (3.5), (3.6), (3.14) and (3.13).
The unitary quarklepton mixing matrices (3.22) make the transformations from lepton
diagonal massmatrices to quark diagonal massmatrices, vice versa.
are ESBgenerated and related to the topquark mass mtsb by the mixing matrices (3.21)
or (3.22). Analogously to eq. (4.1) for the htti, in terms of twofermion 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 leptonfamily index ` is
each of three fermion families (mass eigenstates). In eqs. (5.4){(5.7), the notation h
does not represent new SSBcondensates, but the 1PI functions of fermion mass operator
L R, i.e., the selfenergy functions
f that satisfy the selfconsistent SD equations or
massgap equations.
We use the quarklepton interaction of the third family as an example to show
the quarklepton interactions contribute to the SDequations of fermion selfenergy
functions [97]. The quarklepton 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 quarklepton interactions (5.8) introduce the ESB terms to the SD equations
(massgap equations) for other fermions.
In order to show these ESB terms, we rst approximate the SD equations to be
selfconsistent mass gapequations by neglecting perturbative gauge interactions and using
the large Ncexpansion to the leading order, as indicated by
gure 2. The quarklepton
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 vectorlike coupling and topquark mass mtsb,
see next section 5.2.
lepton and tau neutrino are given by
Corresponding to the tadpole diagrams in
gure 2, the massgap 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 selfconsistent massgap 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 massgap equation for the topquark
mass mtsb generated by the SSB, while eq. (5.11) is just a selfconsistent massgap equation
mass meb and taulepton mass meb are not zero, if the topquark 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 quarklepton interactions (2.11), quark
masses mtsb and mbeb. On the other hand, if the tauneutrino mass meb and taulepton
mass meb are not zero, they also contribute to the selfconsistent massgap equations for
These discussions can be generalized to the threefamily 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 selfconsistent massgap equations
are coupled together.
boson coupling to righthanded fermions
In addition to the ESB terms due to quarklepton interactions, the e ective vertex of
W boson coupling to righthanded fermions [97],
W (p; p0) = i pg2
at the energy scale E , also introduces the ESB terms to the SchwingerDyson equations.
This is the main reason for the nontrivial bottomquark mass mb, once the topquark 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 vectorlike 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 righthanded helicity states of two
highenergy 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 forwardbackward
asymmetry in topquark 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.,
downquark pair to topquark pair, has the tchannel of one SM Wboson exchange. Its
contributions to the asymmetry (5.14) and total ttproduction rate were studied [96] by
assuming a new massive boson W 0 with left and righthanded 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 vectorlike coupling (5.13)
is approximately universal for all fermions [97].
SchwingerDyson equations for fermion selfenergy functions
In order to understand how fermion masses are generated by the ESB and obey their RG
equations, we are bound to study the SchwingerDyson (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 RGequations of these Yukawa couplings
calculated are only valid in the highenergy 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 massshell 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 fourfermion interactions, leading to the mass generation of the second fermion family.
It is worthwhile to mention that at the lowest order (treelevel), SM neutral
gaugebosons ( and Z0) interactions and fourfermion 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 selfenergy functions, as shown in gure 4, to SD equations for fermion selfenergy
functions are negligible.
to the fermion selfenergy 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 massgap 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) familymixing diagram, gure 4 in ref. [97], via
W boson exchange at highenergy scale E ; (ii) Eq. (5.1) via tadpole diagrams gure 2 of
quarklepton interactions (2.11) or (2.12). De ning bare fermion masses
; ; s; c), massgap 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 massgap equations (7.13){(7.16) to the fermion masses
are: (i) the
neutrino and cquark acquire their ESB masses m0 and mc0 from the
topquark mass mt0 via the quarklepton interactions (3.21) between the third and second
families, i.e., M3 and M4; (ii) the squark acquires it ESB mass ms0 via the CKM mixing
and the quarklepton interactions M5; (iii) the lepton acquires its ESB mass m0 via the
PMNS mixing and the quarklepton 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 massgap equations at the scale
eqs. (7.13){(7.16).
On the basis of eqs. (7.15) and (7.18) at the scale
and the cquark massshell
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 squark 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
quarklepton interactions and mixing between the third and second families. The scale
evolution of masses and Yukawa couplings are functions of the topquark one gt( ), see
The rst fermion family
We turn to the masses and Yukawa couplings of the rst fermion family. The coupled SD
gapequations receive ESB contributions from the third and second families, through the
CKM and PMNS mixing as well as quarklepton 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 massgap 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 eneutrino acquires its mass m0 from the tquark
mass mt0 via the quarklepton interaction M7; (ii) the uquark acquires its mass m0u from
the tquark mass mt0 via the quarklepton interaction M8; (iii) the elepton 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 dquark
dominantly acquires its mass m0d from quark masses mu; mc and mt via the CKM mixing,
as well as a small contribution from the quarklepton 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 gapequations 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 dquark 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 fermionfamily mixing due to the
quarklepton interactions and the W boson vectorlike vertex (CKM and PMNS mixing) at high
energies. As a consequence, this leads to the generations of other fermion masses by the
topquark Yukawa coupling gt( ),
gure 1. We approximately analyze the coupled SD
gapequations 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
gapequations, and approximately
nd their masses, consistently with the fermionfamily
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 netuning we
which seems to be consistent with the SM. The topquark mass is generated by the SSB, and
others by the ESB attributed to the topquark mass and familymixing. All masses are calculated
approximately obtain the hierarchy pattern of 12 SMfermion masses, see table 1, and their
Yukawa couplings, consistently with the parameter
w (6.3) and the hierarchy pattern of
9 familymixing parameters Mi.
It is energetically favorable that the SSB solely occurs for the ttchannel (1.1)
generating the topquark mass and three Goldstone modes only. The SSB realizes the approximate
However, this SSB generated vacuum alignment is rearranged 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 gapequations for fermion masses, so that fermions
become massive mt
and fermionfamily mixing matrices in the two ways: (i) the fermionfamilymixing
matrices (3.8) and (3.17) including the CKM and PMNS matrices introduce the ESB terms,
due to the vectorlike coupling
w (5.13) and (6.3) of the W
boson at high energies E
(see preliminary study [32, 33]); (ii) the quarkleptonfamily mixing matrices (3.22)
introduce the ESB terms, due to the quarklepton interactions (3.21) at high energies E . It is
expected that the ESB terms perturbatively rearrange the SSB generated vacuum
alignment, because of the small coupling
w and fermionfamilymixing 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 PMNSmatrix elements already experimentally measured. If
these fermionfamily mixingmatrix 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 fermionfamily
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
wterms 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 fermionfamilymixing 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 UVfundamental 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 topquark mass, are related and determined
upon the chiralsymmetrybreaking ground state of the UVfundamental 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 fermionfamily mixing parameters Mi
10 7) of eqs. (7.5), (7.17) and (7.27). Since we have not understood the
hierarchical mixingmatrix pattern of the UVfundamental 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
massgap equations depend on not only poorly known and totally unknown familymixing
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 massspectra 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 leptonnumber conservation.
Spontaneous symmetry breaking of Ulepton(1) symmetry
In the fourfermion 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
fourfermion operator preserves the global Ulepton(1)symmetry for the leptonnumber
conservation. Similarly to the discussions of the SSB mechanism for the generation of topquark
mass in section 4, the fourfermion 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 leptonnumber is conserved in the ground state (vacuum state) realized
by the SSB of the SM chiral gauge symmetries, whereas the leptonnumber 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 massshell 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
gaugeboson masses MW and MZ experimentally measured, the scale vsterile needs to be
fact, the scale vsterile represents the energy scale of the leptonnumber 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 leptonviolating 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 leptonnumber 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 nontrivial 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 nonrelativistic gauged neutrinos are
Among the three neutrino masssquared 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 nonrelativistic and relativistic sterile neutrinos are given by
Table 2 shows the large mass and masssquared 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 leptonnumber 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 nonrelativstic and relativistic cases. The oscillating lengths read
Lsfg =
Lsfg =
The large values of Majorana mass mfM and masssquared (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 masseigenstates (f 6= f 0) of two
mff0 or masssquared di erences
. Given neutrino
energies, and their masses or masssquared 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 highenergy 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 leptonnumber 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 gaugedsterile 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 groundstate (vacuum) alignment of the e ective theory of relevant fourfermion
operators, the topquark mass is generated by the SSB, and other fermion masses are
originated from the ESB terms, which are induced by the topquark mass via the fermionfamily
mixing, quarklepton interactions and vectorlike W boson coupling at high energies. As
a consequence, the fermion masses are functions of the topquark mass and the fermion
Yukawa couplings are functions of the topquark Yukawa coupling. Based on the approach
adopted and the results obtained in ref. [97], we study the inhomogeneous SDequations
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 fermionfamily
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 quarklepton
mixing matrices (3.22) relates to the quarklepton interactions. These unitary matrices
are composed by the eigenvectors corresponding to eigenvalues (fermionmass spectra) of
fermionmass matrices. They code all information about mixing angles and CPviolations.
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 highorder corrections
and some nonperturbative 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 longstanding problem of fermionmass origin and hierarchy.
Neutrinos and darkmatter 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 masssquared 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 darkmatter (DM) particles. While, the righthanded sterile neutrinos
Dirac type, whose Dirac masses M1D
(see table 2), could be considered as light, weakinteracting \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 leptonnumber nonconservation. What is then the
candidate for light, noninteracting 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 axialvector 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 leptonnumber conservation and lead to the
asymmetry of matter and antimatter. At the end we mention that for strong coupling
G the relevant fourfermion 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 UVstable 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.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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