Further study of the maximally symmetry breaking patterns in an ${\rm SU}(8)$ theory
Ning Chen Zhiyuan Chen Zhanpeng Hou Zhaolong Teng Bin Wang


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thmTheorem
*thm-nonTheorem
*conj[thm-non]Conjecture
*law[thm-non]Law
*ansatz[thm-non]Ansatz
*define[thm-non]Definition
*proof[thm-non]Proof



 

 























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        𝒜ℬ𝒞𝒟ℰℱ𝒢ℋℐ𝒥𝒦ℒℳ𝒩𝒪𝒫𝒬ℛ𝒮𝒯𝒰𝒱𝒲𝒳𝒴𝒵
        𝔄𝔞𝔅𝔟ℭ𝔠𝔇𝔡𝔈𝔢𝔉𝔣𝔊𝔤ℌ𝔥ℑ𝔦𝔍𝔧𝔎𝔨𝔏𝔩𝔐𝔪Ŋ𝔑𝔫𝔒𝔬𝔓𝔭𝔔𝔮ℜ𝔯𝔖𝔰𝔗𝔱𝔘𝔲𝔙𝔳𝔚𝔴𝔛𝔵𝔜𝔶ℨ𝔷
        
        
        
        
         Further study of the maximally symmetry breaking patterns in an SU(8) theory
        
        
        
        
        
        
        
             Ning Chen 0000-0002-0032-9012, Zhiyuan Chen, Zhanpeng Hou, Zhaolong Teng 0000-0002-7141-2331, Bin Wang
            
        School of Physics, Nankai University, Tianjin, 300071, China 
        
        ===========================================================================================================
        
        § ABSTRACT 
        
        
        The SU(8) was previously found to be the minimal simple gauge group where all three-generational Standard Model fermions can be non-trivially embedded, and it is maximally broken into SU(8)→_441≡ SU(4)_s ⊗ SU(4)_W ⊗ U(1)_X_0 at the GUT scale by the SU(8) adjoint Higgs field.
        Gauge symmetries in the strong and the weak sectors are extended by one and two ranks, respectively.
        The sequential strong-weak-weak (SWW) symmetry breaking stages were found to generate the observed hierarchical SM quark/lepton masses as well as the Cabibbo-Kobayashi-Maskawa (CKM) mixing pattern with the precise flavor identifications <cit.>.
        We further study the possible weak-strong-weak (WSW) and weak-weak-strong (WWS) symmetry breaking patterns, and compare with the results that we have obtained by following the SWW sequence.
        The two-loop RGEs following both patterns are derived, where we cannot achieve the gauge coupling unification in the field theory framework.
        Based on these analyses, we suggest the gauge coupling unification to be interpreted in the context of the Kač-Moody Lie algebra.
        
        
        
        Emails:
        <chenning_symmetry@nankai.edu.cn>
        <chenzhiyuan@mail.nankai.edu.cn>
        <houzhanpeng@mail.nankai.edu.cn>
        <tengcl@mail.nankai.edu.cn>
        <wb@mail.nankai.edu.cn>
        empty
        
        
        
        § INTRODUCTION
        
        
        
        The Grand Unified Theory (GUT) was proposed to unify all three fundamental symmetries described by the Standard Model (SM).
        Two original versions that have been widely studied over decades are the (5) Georgi-Glashow theory <cit.> and the (10) Fritzsch-Minkowski theory <cit.>, as well as their supersymmetric extensions <cit.>.
        However, the minimal unified frameworks can not address much on the existing flavor puzzles of all three-generational SM fermions, unless further ingredients are included <cit.>.
        
        
        
        
        
        From the experimental perspective, the indispensable guidance to unveil the flavor puzzle (particularly for quarks and leptons) to date is the discovery <cit.> and the measurements <cit.> of one single SM Higgs boson.
        The LHC measurements have confirmed that the Yukawa couplings at the EW scale between the SM Higgs boson and the (t ,b ,τ ,μ) are consistent with the SM predictions.
        Based on the experimental facts, one natural conjecture is that the hierarchical Yukawa couplings of the SM Higgs boson originate from the flavor non-universalities to all three generations beyond the SM.
        Obviously, the minimal (5) or (10) theories only allows the simply repetitive flavor structure.
        In an earlier study by Georgi <cit.>, he proposed to extend the gauge group beyond the minimal SU(5) so that three-generational SM fermions can be non-trivially embedded.
        Consequently, three-generational SM fermions must transform differently in the UV-complete theory.
        In the original third law of the SU(N) unified theory, Georgi proposed to only include the non-repetitive SU(N) irreps, which lead to a minimal three-generational SU(11) framework with the chiral fermions of
        
        
        { f_L }_ SU(11)^n_g=3   =   
            11 ,4
        _F⊕
            11 , 8
        _F⊕
            11 ,9
        _F⊕
            11 ,10
        _F .
        
        
        In the recent studies, we introduce a concept of chiral irreducible anomaly-free fermion sets (IRAFFSs), which reads <cit.>
        
        
        
        
        
        a chiral IRAFFS is a set of left-handed anti-symmetric fermions of ∑_ m_ _L(), with m_ being the multiplicities of a particular fermion representation of .
        Obviously, the anomaly-free condition reads ∑_ m_  Anom(  _L() ) =0.
        We also require the following conditions to be satisfied for a chiral IRAFFS:
        
        
        
          *  the greatest common divisor (GCD) of the { m_} should satisfy that GCD{  m_} =1;
        
        
        
          *  the fermions in a chiral IRAFFS can no longer be removed, which would otherwise bring non-vanishing gauge anomalies;
        
        
          *  there should not be any singlet, self-conjugate, adjoint fermions, or vectorial fermion pairs in a chiral IRAFFS.
        
        
        
        
        Correspondingly, Georgi's 1979 third law <cit.> can be reformulated as follows <cit.>
        
        
        
        only distinctive chiral IRAFFSs are allowed in the GUT.
        
        
        
        This leads to an SU(8) theory with the minimal chiral fermions of
        
        
        { f_L }_ SU(8)^n_g=3   =   [ 8_F^ω⊕28_F] ⊕[ 8_F^ω̇⊕56_F]  ,  dim_𝐅= 156 ,       Ω≡ ( ω , ω̇)  ,  ω = ( 3 ,  IV ,  V ,  VI)  ,  ω̇= (1̇ , 2̇ , V̇İİ , V̇İİİ , İẊ )  ,
        
        
        with undotted/dotted indices for the 8_F's in the rank-2 chiral IRAFFS and the rank-3 chiral IRAFFS, respectively.
        The Roman numbers and the Arabic numbers are used for the heavy partner fermions and the SM fermions.
        
        
        
        
        
        The SU(8) gauge group has rank seven, which predicts three intermediate symmetry breaking stages between the GUT scale and the EW scale.
        The adjoint Higgs field of 63_H achieves the maximally breaking pattern of SU(8)→_441 through its VEV of
        
        
        ⟨63_H⟩   =   1/ 4  diag(- 𝕀_4× 4 , +𝕀_4× 4 ) v_U  ,
        
        
        according to L.F.Li <cit.>.
        The U(1)_X_0 charges of the SU(8) fundamental representation are defined as follows
        
        
        X̂_0( 8 )    ≡    diag (  - 1/4𝕀_4× 4_4_s ,  + 1/4𝕀_4× 4_4_W ) .
        
        
        In this pattern, the gauge symmetries in the strong and the weak sectors are extended by one and two ranks beyond the SM gauge groups, respectively.
        The sequential symmetry breaking patterns may follow the strong-weak-weak (SWW) <cit.>, the weak-strong-weak (WSW), and the weak-weak-strong (WWS) sequences. 
        These cannot be determined purely by group theory.
        Other different symmetry breaking patterns of the SU(8) group were previously described in Refs. <cit.>.
        
        
        
        
        Through the previous analyses of the SWW symmetry breaking pattern <cit.>, we found (i) one unique SM Higgs boson by looking for the origin of the global U(1)_B-L symmetry, (ii) the hierarchical Yukawa couplings to all three-generational SM quarks/leptons due to the intrinsic symmetry properties, and (iii) the reasonable Cabibbo-Kobayashi-Maskawa (CKM) mixing pattern <cit.> in the quark sector.
        Three intermediate symmetry breaking scales beyond the EW scale enter into the SM quark/lepton mass matrices, and hence are set by requiring a reasonable matching with the experimental measurements.
        Meanwhile, the renormalization group (RG) evolutions of three gauge couplings in the SWW symmetry breaking pattern together with three intermediate scales could not achieve the unification in the field theory context <cit.>.
        In this work, we further analyze two other possible WSW and WWS symmetry breaking patterns of the SU(8) theory, where the extended weak symmetries breaking pattern of _441→_431 [An effective _431 was previously studied in Ref. <cit.>.] happened at the first stage.
        Within each symmetry breaking pattern, we shall look for whether the reported hierarchical SM quark/lepton masses as well as the CKM mixing patterns following the SWW sequence in Ref. <cit.> can be reproduced.
        Furthermore, we shall derive the RG evolutions in both WSW and WWS symmetry breaking patterns to look for the gauge coupling unification.
        
        
        
        
        
        The rest of the paper is organized as follows.
        In Sec. <ref>, we review the SU(8) framework, where three-generational SM fermions are non-trivially embedded, and hence they transform differently in the UV theory. 
        We focus on the WSW and the WWS symmetry breaking patterns, define various gauge U(1) and non-anomalous global U(1)_T charges at different stages, and decompose all SU(8) chiral fermions and Higgs fields accordingly.
        A set of d=5 operators that generate all light SM quark/lepton mass terms other than the top quark mass are collected according to Ref. <cit.>.
        In Secs. <ref> and <ref>, we analyze both WSW and WWS symmetry breaking patterns in details, where we describe the procedures to integrate out the massive vectorlike fermions, and also derive the SM quark/lepton mass terms based on a set of d=5 fermion bi-linear operators and the irreducible Higgs mixing operators.
        In order to reproduce the SM quark/lepton mass hierarchies and the CKM mixing pattern in Ref. <cit.>, the SM flavor identifications are modified for the first and second generations accordingly.
        Accordingly, we obtain two separate benchmark points in both patterns with the suggested intermediate symmetry breaking scales.
        In Sec. <ref>, we obtain the RG evolutions of the minimal SU(8) theory by following both WSW and WWS sequences based on the suggested benchmark points.
        The RG behaviors in neither symmetry breaking patterns are likely to achieve the conventional gauge coupling unification, and their behaviors also match with what we have found in the SWW sequence <cit.> by and large.
        We conclude and make future perspective in Sec. <ref>, where we suggest that the gauge coupling unification in the minimal SU(8) theory to be interpreted in terms of the Kač-Moody Lie algebra.
        In App. <ref>, we collect the SM quark/lepton mass matrices from the SWW symmetry breaking pattern that we have obtained in Refs. <cit.>.
        
        
        
        
        
        
        § THE SU(8) THEORY AND POSSIBLE SYMMETRY BREAKING PATTERNS
        
        
        
         §.§ Overview
        
        The SU(8) theory was formulated by requiring several distinctive chiral IRAFFSs that can lead to three-generational SM fermions at the electroweak scale <cit.>.
        The non-anomalous global DRS symmetries from fermions in Eq. (<ref>) are
        
        
        _ DRS
            SU(8) , n_g=3
           =   [  SU(4)_ω⊗ U(1)_T_2]  ⊗[  SU(5)_ω̇⊗ U(1)_T_3]   ,
        
        
        and we also denote the anomalous global Peccei-Quinn symmetries <cit.> as
        
        
        _ PQ
            SU(8) , n_g=3
           =    U(1)_ PQ_2⊗ U(1)_ PQ_3 .
        
        
        Since the global B-L symmetry should be identical for all three generations, we further require a common non-anomalous U(1)_T≡ U(1)_T_2 =  U(1)_T_3 between two chiral IRAFFSs.
        The anomalous global U(1)_ PQ charges are assigned such that
        
        
                p : q_2  ≠ -3 : +2   ,    p  : q_3 ≠ -3  : +1   .
        
        
        
        
        The most general gauge-invariant Yukawa couplings at least include the following renormalizable and non-renormalizable terms [The term of 56_F56_F28_H  + H.c. vanishes due to the anti-symmetric property <cit.>. Instead, only a d=5 non-renormalizable term of 1/ M_ pl56_F56_F28_H_ ,ω̇^†63_H is possible to generate masses for vectorlike fermions in the 56_F. Since it transforms as an SU(5)_ω̇ vector and carries non-vanishing U(1)_ PQ charge of p+3q_3 ≠ 0 from Eq. (<ref>), it is only possible due to the gravitational effect.]
        
        
        -_Y   =    Y_8_F^ω28_F8_H_ ,ω  +  Y_28_F28_F70_H   +     Y_8_F^ω̇56_F28_H_ ,ω̇   +  c_4 / M_ pl56_F56_F28_H_ ,ω̇^†63_H + H.c. ,
        
        
        with the reduced Planck scale of M_ pl= ( 8 π G_N)^-1/2= 2.4 × 10^18  GeV.
        All renormalizable Yukawa couplings and non-renormalizable Wilson coefficients are expected to be (Y_ , Y_ , Y_ , c_4 )∼(1).
        Altogether, we collect the SU(8) Higgs fields as follows
        
        
        { H }_ SU(8)^n_g=3   =   8_H_ , ω⊕28_H_ , ω̇⊕70_H⊕63_H ,  dim_𝐇= 547  ,
        
        
        where the adjoint Higgs field of 63_H is real while all others are complex.
        Accordingly, the non-anomalous global U(1)_T charges and the anomalous global U(1)_ PQ charges for all fermions and Higgs fields are assigned in Tab. <ref>.
        
        
        
        
        
        
        
        
        
         §.§ The SWW symmetry breaking pattern
        
        The SWW symmetry breaking pattern of the SU(8) theory follows the sequence of
        
        
        
               SU(8) _441_341_331_ SM SU(3)_c⊗ U(1)_ EM , 
              _441≡ SU(4)_s⊗ SU(4)_W ⊗ U(1)_X_0  ,  _341≡ SU(3)_c⊗ SU(4)_W ⊗ U(1)_X_1  ,
              _331≡ SU(3)_c⊗ SU(3)_W ⊗ U(1)_X_2  , _ SM≡ SU(3)_c⊗ SU(2)_W ⊗ U(1)_Y  ,
              with   v_U≫ v_441≫ v_341≫ v_331≫ v_ EW ,
        
        
        which was previously studied in Refs. <cit.>.
        Once one assumes that the extended gauge symmetries in the strong sector break first, there is no ambiguity of the sequential symmetry breaking patterns.
        Sequentially, the U(1)_X_1, U(1)_X_2, and U(1)_Y charges are defined according to the SU(4)_s and the SU(4)_W fundamental representations as follows
        
        
        X̂_1(4_s)    ≡    diag (   (- 1/12+ _0 ) 𝕀_3× 3_3_c , 1/4+ _0 )  ,
        
        X̂_2 ( 4_W )   ≡    diag (   ( 1/12 + _1 ) 𝕀_3× 3_3_W , -1/4 + _1 )   , 
        
        Ŷ ( 4_W )   ≡    diag (  ( 1/6+ _2 ) 𝕀_2× 2 ,- 1/3+ _2  , _2 )    =    diag (   ( 1/4 + _1 ) 𝕀_2× 2_2_W ,  ( - 1/4 + _1  ) 𝕀_2× 2)  , 
        
        Q̂_e ( 4_W )   ≡    T_ SU(4) ^3 +  Ŷ ( 4_W ) =  diag ( 3/4 + _1   , ( - 1/4 + _1  ) 𝕀_3× 3)  . 
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        The non-anomalous global U(1)_T symmetry becomes the global U(1)_B-L at the EW scale according to the following sequence <cit.>
        
              _441 : ^'≡ - 4t _0  ,   _341 : ^''≡^' + 8t_1  , 
              _331 : ^'''≡^'' ,   _ SM : - ≡^''' .
        
        
        
         §.§ The WSW symmetry breaking pattern
        
        The WSW symmetry breaking pattern of the SU(8) theory follows the sequence of
        
        
               SU(8) _441_431_331_ SM SU(3)_c⊗ U(1)_ EM , 
              _441≡ SU(4)_s⊗ SU(4)_W ⊗ U(1)_X_0  ,  _431≡ SU(4)_c⊗ SU(3)_W ⊗ U(1)_X_1  ,
              _331≡ SU(3)_c⊗ SU(3)_W ⊗ U(1)_X_2  , _ SM≡ SU(3)_c⊗ SU(2)_W ⊗ U(1)_Y  ,
              with   v_U≫ v_441≫ v_431≫ v_331≫ v_ EW .
        
        
        The U(1)_X_1, U(1)_X_2, and U(1)_Y charges along the WSW sequence are defined according to the SU(4)_s and the SU(4)_W fundamental representations as follows
        
        
        X̂_1 ( 4_W )   ≡    diag (   ( 1/12 + _0 ) 𝕀_3× 3_3_W , -1/4 + _0 )   , 
        
        X̂_2(4_s)    ≡    diag (   (- 1/12+ _1 ) 𝕀_3× 3_3_c , 1/4+ _1 )  ,
        
        Ŷ ( 4_W )   ≡    diag (  ( 1/6+ _2 ) 𝕀_2× 2 ,- 1/3+ _2  , _2 )    =    diag (   ( 1/4 + _0 ) 𝕀_2× 2_2_W ,  ( - 1/4 + _0  ) 𝕀_2× 2)  , 
        
        Q̂_e ( 4_W )   ≡    T_ SU(4) ^3 +  Ŷ ( 4_W ) =  diag ( 3/4 + _0   , ( - 1/4 + _0  ) 𝕀_3× 3)  . 
        
        The non-anomalous global U(1)_T symmetry becomes the global U(1)_B-L at the EW scale according to the following sequence <cit.>
        
              _441 : ^'≡ + 4t _0  ,   _431 : ^''≡^' -8t_1  , 
              _331 : ^'''≡^'' + 8 t _2  ,   _ SM : - ≡^''' .
        
        
        
        
        By following the symmetry breaking pattern in Eq. (<ref>), we tabulate the fermion representations at various stages of the SU(8) theory in Tabs. <ref>, <ref>, and <ref>.
        
        
        
        For the right-handed down-type quarks of _R^Ω^c, they are named as follows
        
        
              _R^1̇^c ≡d_R^c  ,  _R^2̇^c ≡s_R^c  , _R^V̇İİ^c ≡_R^'''''^c  ,  _R^V̇İİİ^c ≡_R^'''^c   , _R^İẊ^c ≡_R^''''^c  , 
              _R^3^c ≡b_R^c  , _R^ IV ^c ≡_R^''^c  ,  _R^ V ^c ≡_R^c   , _R^ VI ^c ≡_R^'^c   .
        
        
        For the left-handed SU(2)_W lepton doublets of (_L^Ω , - _L^Ω ), they are named as follows
        
        
                 ( _L^1̇ ,  - _L^1̇)  ≡ (e_L , - ν_e L )  ,  ( _L^2̇ ,   - _L^2̇)  ≡( μ_L  , - ν_μ L )   , 
                ( _L^V̇İİ ,  - _L^V̇İİ)  ≡ ( _L^'''' , - _L^'''' )   ,   ( _L^V̇İİİ , - _L^V̇İİİ  )  ≡ ( _L^''' , - _L^''' )  ,   ( _L^İẊ ,  - _L^İẊ ) ≡ ( _L^''''' ,  - _L^''''' )   ,
                ( _L^ 3 , - _L^3) ≡ ( τ_L  , - ν_τ L) ,   ( _L^ IV  , - _L^ IV ) ≡ ( _L  , - _L ) , 
               ( _L^ V  , -  _L^ V ) ≡ ( _L^'' , - _L^'' )   ,  ( _L^ VI  , - _L^ VI  ) ≡  ( _L^' , - _L^' )  .
        
        
        Through the analysis in Sec. <ref>, we shall see that all heavy (^Ω , ^Ω , ^Ω) (with Ω= IV , … ,İẊ) acquire vectorlike masses during the intermediate symmetry breaking stages.
        For the remaining left-handed sterile neutrinos of (  _L^Ω , _L^Ω^' , _L^Ω^'' ), several of them are massive and they are named as follows
        
        
              _L^ IV≡_L^'' , _L^ IV^'≡_L^ ,  _L^ V^'≡_L^' , _L^V̇İİ^'≡_L^''' .
        
        
        Notice that, the first- and second-generational SM fermions are named differently in Tab. <ref> as compared to the names following the SWW symmetry breaking pattern.
        This will be elaborated in our derivation of the SM quark/lepton mass matrices in Sec. <ref>.
        
        
        
        
        
        We decompose the Higgs fields in Yukawa term into components that can be responsible for the sequential symmetry breaking pattern in Eq. (<ref>).
        All possible Higgs components that are likely to develop VEVs for the corresponding symmetry breaking stages are marked by ⟨ ... ⟩, while their UV origins are denoted by underlines.
        For Higgs fields of 8_H_ , ω they read
        
        
        
            8_H_ ,ω   ⊃     ( 4 , 1 , +1/4 )_𝐇 , ω⊕⟨ ( 1 , 4 , -1/4 )_𝐇 , ω⟩
               ⊃   ⟨ ( 4 , 1 , +1/4  )_𝐇 , ω⟩⊕ ( 1 , 3 , -1/3  )_𝐇 , ω
               ⊃   ⟨  ( 1 , 3 , -1/3 )_𝐇 , ω⟩⊃⟨ ( 1 , 2 , -1/2 )_𝐇 , ω⟩ .
        
        
        For Higgs fields of 28_H_ ,ω̇, they read
        
        
        
            28_H_ ,ω̇   ⊃    ( 6 , 1 ,  +1/2 )_𝐇 , ω̇⊕ ( 1 , 6 , -1/2 )_𝐇 , ω̇⊕ ( 4 , 4 , 0 )_𝐇 , ω̇
               ⊃   [   ( 1 , 3 , -1/3 )_𝐇 , ω̇^'⊕ ( 1 , 3 , -2/3 )_𝐇 , ω̇] ⊕ [   ( 4 , 3 , -1/12 )_𝐇 , ω̇⊕⟨ ( 4 , 1 , +1/4 )_𝐇 , ω̇⟩ ] 
               ⊃    [ ⟨ ( 1 , 3 , -1/3 )_𝐇 , ω̇^'⟩⊕ ( 1 , 3 , -2/3 )_𝐇 , ω̇ ] ⊕⟨ ( 1 , 3 , -1/3 )_𝐇 , ω̇⟩
               ⊃    [ ⟨ ( 1 , 2 , -1/2 )_𝐇 , ω̇^'⟩⊕⟨ ( 1 , 2 , -1/2 )_𝐇 , ω̇⟩ ] ⊕⟨ ( 1 , 2 , -1/2 )_𝐇 , ω̇⟩ .
        
        
        
        For Higgs field of 70_H, they read
        
        
        
            70_H   ⊃    ( 1 , 1  , -1 )_𝐇^''⊕ ( 1 , 1  , +1 )_𝐇^''''⊕  ( 4 , 4 , -1/2 )_𝐇⊕ ( 6  , 6  , 0 )_𝐇⊕ ( 4 , 4 , +1/2 )_𝐇
               ⊃    ( 4 , 3 , +5/12 )_𝐇⊃ ( 1 , 3 , +2/3 )_𝐇^'''⊃⟨ ( 1 , 2 , +1/2 )_𝐇^'''⟩ .
        
        
        
        
        
        Schematically, we assign the Higgs VEVs according to the decompositions in Eqs. (<ref>), (<ref>), and (<ref>) as follows
        
        
        _441→_431    :    ⟨ ( 1 , 4 , -1/4 )_𝐇 ,  IV⟩≡1/√(2)W_4 ,  IV , 
        
        _431→_331    :    ⟨ ( 4 , 1 , +1/4 )_𝐇 ,  V⟩≡1/√(2) w_4 ,  V ,  ⟨ ( 4 , 1 , +1/4 )_𝐇 ,1̇,  V̇İİ⟩≡1/√(2)  w_4 , 1̇,V̇İİ ,
        
        _331→_ SM    :    ⟨ ( 1 , 3 , -1/3 )_𝐇 , 3,VI⟩≡1/√(2) V_3 , 3,VI , 
              ⟨ ( 1 , 3 , -1/3 )_𝐇  ,İẊ⟩≡1/√(2) V_3 ,İẊ ,  ⟨ ( 1 , 3 , -1/3 )_𝐇 ,2̇, V̇İİİ^'⟩≡1/√(2) V_3 , 2̇,V̇İİİ^' ,  
        
          EWSB    :    ⟨ ( 1 , 2 , +1/2 )_𝐇^'''⟩≡1/√(2) v_ EW .
        
        For our later convenience, we also parametrize different symmetry breaking VEVs
        
        
        
            ζ_0 ≡ v_U / M_ pl ,   ζ_1 ≡ W_4 ,  IV/ M_ pl ,   ζ_2 ≡ w_4 ,  V/ M_ pl ,    ζ̇_2 ≡ w_4 ,1̇,V̇İİ/ M_ pl ,   
                  ζ_3 ≡ V_3 ,3, VI/ M_ pl ,   ζ̇_3^'≡ V_3 , 2̇,V̇İİİ^'/ M_ pl ,   ζ̇_3 ≡ V_3 , İẊ/ M_ pl , 
                  ζ_0 ≫ζ_1 ≫ζ_2 ∼ζ̇_2  ≫ζ_3 ∼ζ̇_3^'∼ζ̇_3   ,   ζ_i j≡ζ_j /ζ_i  ,    ( i < j )  ,
        
        
        in terms of dimensionless quantities.
        Here, we adopt the conventions as follows
        
        
        
          *  The notations of (W , w , V) and the dimensionless quantities of ( ζ_1 , ζ_2 ∼ζ̇_2 , ζ_3 ∼ζ̇_3^'∼ζ̇_3) are used for the Higgs VEVs at the first, second, and the third symmetry breaking stages, regardless of the specific symmetry breaking patterns.
         
         
          * W_4 and w_4 represent the VEVs for the _441→_431 and the _431→_331 stages in Eqs. (<ref>).
         In the SWW sequence <cit.>, W_4 and w_4 represent the VEVs for the _441→_341 and the _341→_331 stages instead.
         
        
        
        
         §.§ The WWS symmetry breaking pattern
        
        The WWS symmetry breaking pattern of the SU(8) theory follows the sequence of
        
        
               SU(8) _441_431_421_ SM SU(3)_c⊗ U(1)_ EM , 
              _441≡ SU(4)_s⊗ SU(4)_W ⊗ U(1)_X_0  ,  _431≡ SU(4)_s⊗ SU(3)_W ⊗ U(1)_X_1  ,
              _421≡ SU(4)_s⊗ SU(2)_W ⊗ U(1)_X_2  , _ SM≡ SU(3)_c⊗ SU(2)_W ⊗ U(1)_Y  ,
              with   v_U≫ v_441≫ v_431≫ v_331≫ v_ EW .
        
        
        The U(1)_X_1, U(1)_X_2, and U(1)_Y charges along the WWS sequence are defined according to the SU(4)_s and the SU(4)_W fundamental representations as follows
        
        
        X̂_1 ( 4_W )   ≡    diag (   ( 1/12 + _0 ) 𝕀_3× 3_3_W , -1/4 + _0 )   , 
        
        X̂_2(3_W)    ≡    diag (   ( 1/6 + _1 ) 𝕀_2 × 2_2_W  , - 1/3+ _1 )  ,
        
        Ŷ ( 4_s )   ≡    diag (    (- 1/12+ _2 ) 𝕀_3× 3_3_c , 1/4+ _2 )  .
        
        
        
        The non-anomalous global U(1)_T symmetry becomes the global U(1)_B-L at the EW scale according to the following sequence <cit.>
        
              _441 : ^'≡ + 4t _0  ,   _431 : ^''≡^' , 
              _421 : ^'''≡^'' - 8 t _2  ,   _ SM : - ≡^''' + 8 t  .
        
        
        
        
        
        By following the symmetry breaking pattern in Eq. (<ref>), we tabulate the fermion representations at various stages of the SU(8) theory in Tabs. <ref>, <ref>, and <ref>.
        For the right-handed down-type quarks of _R^Ω^c, they are named as follows
        
        
              _R^1̇^c ≡d_R^c  ,  _R^2̇^c ≡s_R^c  , _R^V̇İİ^c ≡_R^'''''^c  ,  _R^V̇İİİ^c ≡_R^'''^c   , _R^İẊ^c ≡_R^''''^c  , 
              _R^3^c ≡b_R^c  , _R^ IV ^c ≡_R^''^c  ,  _R^ V ^c ≡_R^^c   , _R^ VI ^c ≡_R^'^c   .
        
        
        For the left-handed SU(2)_W lepton doublets of (_L^Ω , - _L^Ω ), they are named as follows
        
        
                 ( _L^1̇ ,  - _L^1̇)  ≡ (e_L , - ν_e L )  ,  ( _L^2̇ ,   - _L^2̇)  ≡( μ_L  , - ν_μ L )   , 
                ( _L^V̇İİ ,  - _L^V̇İİ)  ≡ ( _L^'''' , - _L^'''' )   ,   ( _L^V̇İİİ , - _L^V̇İİİ  )  ≡ ( _L^''' , - _L^''' )  ,   ( _L^İẊ ,  - _L^İẊ ) ≡ ( _L^''''' ,  - _L^''''' )   ,
                ( _L^ 3 , - _L^3) ≡ ( τ_L  , - ν_τ L) ,   ( _L^ IV  , - _L^ IV ) ≡ ( _L  , - _L ) , 
               ( _L^ V  , -  _L^ V ) ≡ ( _L^'' , - _L^'' )   ,  ( _L^ VI  , - _L^ VI  ) ≡  ( _L^' , - _L^' )  .
        
        
        Through the analysis in Sec. <ref>, we shall see that all heavy (^Ω , ^Ω , ^Ω) (with Ω= IV , … ,İẊ) acquire vectorlike masses during the intermediate symmetry breaking stages.
        For the remaining left-handed sterile neutrinos of (  _L^Ω , _L^Ω^' , _L^Ω^'' ), several of them are massive and they are named as follows
        
        
              _L^ IV≡_L^'' , _L^ IV^'≡_L^ ,  _L^ V^'≡_L^' , _L^V̇İİ^'≡_L^''' .
        
        
        Similar to the situation in the WSW pattern, the SM fermion names in Tab. <ref> are exchanged from what we have obtained in the SWW pattern.
        The detailed derivation of the corresponding SM quark/lepton mass matrices will be elaborated in Sec. <ref>.
        
        
        
        
        
        We decompose the Higgs fields in Yukawa term into components that can be responsible for the sequential symmetry breaking pattern in Eq. (<ref>).
        For Higgs fields of 8_H_ , ω they read
        
        
        
            8_H_ ,ω   ⊃     ( 4 , 1 , +1/4 )_𝐇 , ω⊕⟨ ( 1 , 4 , -1/4 )_𝐇 , ω⟩
               ⊃    ( 4 , 1 , +1/4 )_𝐇 , ω⊕⟨ ( 1 , 3 , -1/3 )_𝐇 , ω⟩
               ⊃   ⟨ ( 4 , 1 , +1/4 )_𝐇 , ω⟩⊕ ( 1 , 2 , -1/2 )_𝐇 , ω⊃⟨ ( 1 , 2 , -1/2 )_𝐇 , ω⟩ .
        
        
        For Higgs fields of 28_H_ ,ω̇, they read
        
        
        
            28_H_ ,ω̇   ⊃    ( 6 , 1 ,  +1/2 )_𝐇 , ω̇⊕ ( 1 , 6 , -1/2 )_𝐇 , ω̇⊕ ( 4 , 4 , 0 )_𝐇 , ω̇
               ⊃   
                ⟨  ( 1 , 3 , -1/3 )_𝐇 , ω̇^'⟩⊕ ( 1 , 3 , -2/3 )_𝐇 , ω̇
            ⊕
                ( 4 , 3 , -1/12 )_𝐇 , ω̇⊕ ( 4 , 1 , +1/4 )_𝐇 , ω̇^'
            
               ⊃    [  ( 1 , 2 , -1/2 )_𝐇 , ω̇^'⊕ ( 1 , 2 , -1/2 )_𝐇 , ω̇ ]       ⊕
                ( 4 , 2 , -1/4 )_𝐇 , ω̇⊕⟨ ( 4 , 1 , +1/4 )_𝐇 , ω̇⟩⊕⟨ ( 4 , 1 , +1/4 )_𝐇 , ω̇^'⟩
            
               ⊃   
                ⟨ ( 1 , 2 , -1/2 )_𝐇 , ω̇^'⟩⊕⟨ ( 1 , 2 , -1/2 )_𝐇 , ω̇⟩
            ⊕
                ⟨ ( 1 , 2 , +1/2 )_𝐇 , ω̇⟩
             .
        
        
        For Higgs field of 70_H, they read
        
        
        
            70_H   ⊃    ( 1 , 1  , -1 )_𝐇^''⊕ ( 1 , 1  , +1 )_𝐇^''''⊕ ( 4 , 4 , +1/2 )_𝐇⊕  ( 4 , 4 , -1/2 )_𝐇⊕ ( 6  , 6  , 0 )_𝐇
               ⊃    ( 4 , 3 , +5/12 )_𝐇⊃ ( 4 , 2 , +1/4 )_𝐇^⊃⟨ ( 1 , 2 , +1/2 )_𝐇^'''⟩ .
        
        
        
        
        
        Schematically, we assign the Higgs VEVs according to the symmetry breaking pattern as follows
        
        
        _441→_431    :    ⟨ ( 1 , 4 , -1/4 )_𝐇 ,  IV⟩≡1/√(2)W_4 ,  IV , 
        
        _431→_421    :    ⟨ ( 1 , 3 , -1/3 )_𝐇 , V⟩≡1/√(2) w_3 ,  V ,  ⟨ ( 1 , 3 , -1/3 )_𝐇 , 1̇, V̇İİ^'⟩≡1/√(2)  w_3 , 1̇,V̇İİ ,
        
        _421→_ SM    :    ⟨ ( 4 , 1 , +1/4 )_𝐇 ,3, VI⟩≡1/√(2) V_4 ,3, VI ,       ⟨ ( 4 , 1 , +1/4 )_𝐇 , İẊ⟩≡1/√(2) V_4 ,İẊ ,  ⟨ ( 4 , 1 , +1/4 )_𝐇 ,2̇ , V̇İİİ^'⟩≡1/√(2) V_4 ,2̇ , V̇İİİ^' ,  
        
          EWSB    :    ⟨ ( 1 , 2 , +1/2 )_𝐇^'''⟩≡1/√(2) v_ EW .
        
        At the third stage, the Higgs VEVs from the rank-3 sector are due to the components of ( 4 , 1 , +1/4 )_𝐇 , İẊ⊂ ( 4 , 4 , 0 )_𝐇 , İẊ and ( 4 , 1 , +1/4 )_𝐇 ,2̇ , V̇İİİ^'⊂ ( 4 , 4 , 0 )_𝐇 , 2̇ , V̇İİİ.
        This is different from the third stage of the WSW pattern, where the Higgs VEVs from the rank-3 sector are from different irreps of ( 1 , 6 , -1/2 )_𝐇 , 2̇ , V̇İİİ and ( 4 , 4 , 0 )_𝐇 ,  İẊ in Eq. (<ref>), respectively.
        For our later convenience, we also parametrize different symmetry breaking VEVs
        
        
        
            ζ_0 ≡ v_U / M_ pl ,   ζ_1 ≡ W_4 ,  IV/ M_ pl ,   ζ_2 ≡ w_3 , 3, V/ M_ pl ,    ζ̇_2 ≡ w_3 ,1̇,V̇İİ/ M_ pl ,   
                  ζ_3 ≡ V_4 , VI/ M_ pl ,   ζ̇_3^'≡ V_4 ,2̇ , V̇İİİ^'/ M_ pl ,   ζ̇_3 ≡ V_4 , İẊ/ M_ pl , 
                  ζ_0 ≫ζ_1 ≫ζ_2 ∼ζ̇_2  ≫ζ_3 ∼ζ̇_3^'∼ζ̇_3   ,   ζ_i j≡ζ_j /ζ_i  ,    ( i < j )  ,
        
        
        in terms of dimensionless quantities.
        Notice that we use the same notations as in Eq. (<ref>) to manifest the intrinsic hierarchies, while their symmetry breaking patterns should be distinguishable.
        
        
        
        
        
        
         §.§ The d=5 operators for the SM quark and lepton masses
        
        
        In Ref. <cit.>, we found that only the top quark obtains the natural tree-level mass from the Y_28_F28_F70_H + H.c. term with Y_∼(1) in Eq. (<ref>).
        All other lighter SM quark/lepton masses are due to the d=5 operators <cit.>, which all explicitly break the emergent global symmetries in Eqs. (<ref>) and (<ref>).
        Among them, we have the d=5 direct Yukawa coupling terms of
        
        
        
        c_3 _^ (3 ,2)   =    c_3 8_F^ω̇56_F·28_H_ , κ̇^†·70_H^† ,
        
        c_4 _^ (4 ,1)   =    c_4 56_F56_F·28_H_ , ω̇·70_H ,
        
        c_5 _^ (5 ,1)   =    c_5 28_F56_F·8_H_ ,ω·70_H .
        
        The operators of _^ (4 ,1) and _^ (5 ,1) were found to generate the hierarchical up-type quark masses.
        To generate the masses for all down-type quarks and charged leptons, we conjecture two sets of Higgs mixing terms
        
        
        
        d_𝒜 _𝒜^d=5   ≡   d_𝒜 ϵ_ω_1 ω_2 ω_3  ω_4 8_H_ , ω_1^†8_H_ , ω_2^†8_H_ , ω_3^†8_H_ , ω_4 ^†70_H^† ,    = 2 ( 2p + 3 q_2 ) ≠ 0  , 
        
        d_ℬ _ℬ^d=5   ≡   d_ℬ    ( 28_H_ ,κ̇_1 ^†28_H_ ,κ̇_2  ) ·28_H_ ,ω̇_1^†28_H_ ,ω̇_2^†70_H^† ,    =  2 ( p + q_2 + q_3)  ,        with  κ̇_2 ≠ ( κ̇_1  , ω̇_1  , ω̇_2 )   , 
        
        where one of the Higgs fields will mix with two renormalizable Yukawa couplings of Y_8_F^ω28_F8_H_ ,ω+ Y_8_F^ω̇56_F28_H_ ,ω̇   + H.c. in Eq. (<ref>), as were depicted in Fig. <ref>.
        Below, we will re-derive the SM quark/lepton mass terms along two separate WSW and WWS symmetry breaking patterns.
        
        
        
        
        
        
        
        § THE WSW SYMMETRY BREAKING PATTERN
        
        
        
         §.§ The first stage
        
        The first symmetry breaking stage of _441→_431 is achieved by ( 1 , 4 ,  -1/4 )_H ,ω⊂8_H_ , ω in the rank-2 sector, according to their _431-invariant and U(1)_ T^'-neutral components in Tab. <ref>.
        
        
        	
        The Yukawa coupling between 8_F^ω and 28_F and the corresponding vectorlike fermion masses can be expressed as
        
        
                Y_8_F^ω28_F8_H_ ,ω + H.c. 
               ⊃    Y_[ ( 4 , 1 , +1/4)_𝐅^ω⊗ ( 4 , 4 , 0 )_𝐅⊕ ( 1 , 4 , -1/4 )_𝐅^ω⊗ ( 1 , 6 , +1/2 )_𝐅] ⊗⟨ ( 1 , 4 , -1/4)_𝐇 ,ω⟩ + H.c. 
           
            
               ⇒   1/√(2)Y_( _L^''_R^''^c + _L^''_R^'' c + _L _R ^c - _L  _R ^c+_L _R^c )W_4, IV + H.c.   .
        
        
        Without loss of generality, we choose ω =  IV according to the VEV assignment in Eq. (<ref>) at this stage.
        Thus, we can identify that ( 3 , 1 , +1/3 )_𝐅^ IV ≡_R^''^c  , ( 1 , 2 , -1/2 )_𝐅^ IV ≡ ( _L  , - _L ), and (_L^ IV , _L^ IV^' )  ≡ ( _L^'' , _L).
        
        
        
        
        
        After this stage, the remaining massless fermions expressed in terms of the _431 irreps are the following
        
        
        
            ( 4 , 1 , +1/4 )_𝐅^Ω⊕[ ( 1 , 3 , -1/3 )_𝐅^Ω⊕ ( 1 , 1 , 0 )_𝐅^Ω^'']  ⊕⊂8_F^Ω ,       Ω = ( ω , ω̇)  ,  ω = (3 ,  V ,  VI )  ,  ω̇= ( 1̇ , 2̇ , V̇İİ ,V̇İİİ , İẊ)  ,       ( 1 , 1 , 0 )_𝐅^ IV^''⊂8_F^ IV ,
                  ( 6 , 1 , -1/2 )_𝐅⊕[  ( 1 , 3 , +1/3 )_𝐅⊕ ( 1 , 3 , +2/3 )_𝐅] ⊕[ ( 4 , 3 , +1/12 )_𝐅⊕ ( 4 , 1 , -1/4 )_𝐅] ⊂28_F , 
                  [ ( 1 , 3 , +2/3 )_𝐅^'⊕ ( 1 , 1 , -1 )_𝐅^''] ⊕ ( 4 , 1 , -3/4 )_𝐅⊕[ ( 4 , 3 , +1/12 )_𝐅⊕ ( 4 , 3 , +5/12)_𝐅]    ⊕   [ ( 6 , 3 , -1/6 )_𝐅⊕ ( 6 , 1 , -1/2)_𝐅^']   ⊂56_F .
        
        
        Fermions that become massive at this stage are crossed out by slashes.
        Loosely speaking, we find only one of the massive 8_F^Ω is integrated out from the anomaly-free conditions of [  SU(4)_s]^2 · U(1)_X_1=0, SU(3)_W
        ^2 · U(1)_X_1=0, and U(1)_X_1
        ^3=0, except for one left-handed sterile neutrino of _L^ IV^''≡ ( 1 , 1 , 0 )_F^ IV^''⊂8_F^ IV.
        
        
        
        
        
         §.§ The second stage
        
        The second symmetry breaking stage can be achieved by ( 4 , 1 ,  +1/4 )_H ,ω⊂8_H_ , ω in the rank-2 sector and ( 4 , 1 ,  +1/4 )_H , ω̇⊂28_H_ , ω̇ in the rank-3 sector, according to their _331-invariant and U(1)_ T^''-neutral components in Tab. <ref>.
        
        
        
        	
        	
        	
        The Yukawa coupling between 8_F^ω and 28_F and the corresponding vectorlike fermion masses can be expressed as
        
        
        
            Y_8_F^ω28_F8_H_ ,ω + H.c. 
                   ⊃       Y_[ ( 4 , 1 , +1/4)_𝐅^ω⊗ ( 6 , 1 , -1/2 )_𝐅⊕  ( 1 , 4 , -1/4)_𝐅^ω⊗ ( 4 , 4 , 0)_𝐅]⊗ ( 4 , 1 , +1/4 )_𝐇 ,ω+ H.c. 
                   ⊃   
                Y_[ ( 4 , 1 , +1/4)_𝐅^ω⊗ ( 6 , 1 , -1/2 )_𝐅⊕  ( 1 , 3 , -1/3)_𝐅^ω⊗ ( 4 , 3 , +1/12)_𝐅   ⊕    ( 1 , 1 , 0)_𝐅^ω^''⊗ ( 4 , 1 , -1/4)_𝐅] ⊗⟨ ( 4 , 1 , +1/4 )_𝐇 ,ω⟩  + H.c. 
             
            
            
                   ⇒   1/√(2)Y_( _L _R^c +_L^''_R^''^c  -_L^''_R^''^c+_L^' c_R^' c  + _L^V^''_R^'' c) w_4,  V + H.c.   .
        
        
        Without loss of generality, we choose ω = V at this stage. 
        Thus, we can identify that ( 3 , 1 , +1/3 )_𝐅^ V ≡_R^c  , ( 1 , 2 , -1/2 )_𝐅^ V ≡ ( _L^'' , - _L^'' ), and _L^ V^'≡_L^'.
        
        
        
        
        The Yukawa coupling between 8_F^ω̇ and 56_F and the corresponding vectorlike fermion masses can be expressed as
        
        
        
            Y_8_F^ω̇56_F28_H_ ,ω̇ + H.c. 
               ⊃    Y_[ ( 4 , 1 , +1/4)_𝐅^ω̇⊗   ( 6 , 4 , -1/4)_𝐅⊕  ( 1 , 4 , -1/4 )_𝐅^ω̇⊗ ( 4 , 6 , +1/4 )_𝐅]  ⊗  ( 4 , 4 , 0)_𝐇 , ω̇ + H.c. 
               ⊃    Y_[ ( 4 , 1 , +1/4)_𝐅^ω̇⊗   ( 6 , 1 , -1/2)_𝐅^'⊕  ( 1 , 3 , -1/3 )_𝐅^ω̇⊗ ( 4 , 3 , +1/12 )_𝐅]  ⊗⟨ ( 4 , 1 , +1/4)_𝐇 , ω̇⟩ + H.c. 
            
            
               ⇒   1/√(2) Y_(  _L^'''''_R^'''''^c+_L^''''_R^''''^c -   _L^''''_R^''''^c + _L^'''_R^''' c ) w_4 , V̇İİ   +   1/√(2) Y_(  _L^'''''d_R ^c+e_L  _R^''''^c -   ν_e   L_R^''''^c + _L^1̇_R^''' c ) w_4 , 1̇ + H.c.  .
        
        
        Without loss of generality, we choose ω̇= ( 1̇ , V̇İİ ) according to the VEV assignment in Eq. (<ref>) at this stage. 
        Thus, we can identify that ( 3 , 1 , +1/3 )_𝐅^V̇İİ≡_R^'''''^c, ( 1 , 2 , -1/2 )_𝐅^V̇İİ≡ ( _L^'''' , - _L^'''' ), and _L^V̇İİ^'≡_L^'''.
        
        
        
        
        The (10_F , 10_F)-pair within the 56_F can obtain the vectorlike masses through the following d=5 operator
        
        
        
            c_4/M_pl56_F56_F⟨63_H⟩28_H_ ,ω̇^† + H.c. 
                   ⊃   c_4   v_U /M_pl[   (  1 , 4  , +3/4 )_𝐅⊗  ( 4 , 1 , -3/4 )_𝐅⊕ (  4 , 6 , +1 /4    )_𝐅⊗  ( 6 , 4  , -1/4  )_𝐅] ⊗ ( 4 , 4 , 0 )_𝐇 , ω̇^†  + H.c. 
                   ⊃   c_4   v_U /M_pl[ (  1 , 1 , +1 )_𝐅^''⊗  ( 4 , 1 , -3/4 )_𝐅⊕ (  4 , 3 , +5/12 )_𝐅⊗  ( 6 , 3 , -1/6 )_𝐅] ⊗⟨ ( 4 , 1 , +1/4)_𝐇 , ω̇^†⟩ + H.c. 
            
                
                   ⇒    c_4 /√(2)ζ_0 ( _L _R^c+_L  _R^c - _L  _R^c+_L _R^c  ) w_4 , 1̇ , V̇İİ + H.c.  .
        
        
        After integrating out the massive fermions, the remaining massless fermions expressed in terms of the _331 irreps are the following
        
        
        
            [ ( 3 , 1 , +1/3 )_𝐅^Ω⊕ ( 1 , 1 , 0 )_𝐅^Ω]  ⊕[ ( 1 , 3 , -1/3 )_𝐅^Ω⊕ ( 1 , 1 , 0 )_𝐅^Ω^'']  ⊂8_F^Ω ,       Ω = ( ω , ω̇)  ,   ω = (3 ,  VI) ,    ω̇= (1̇ , 2̇ , V̇İİİ , İẊ ) ,       ( 1 , 1 , 0)_𝐅^ IV^''⊂8_F^ IV ,    ( 1 , 1 , 0)_𝐅^ V⊕ ( 1 , 1 , 0)_𝐅^ V^''⊂8_F^ V  ,          ( 1 , 1 , 0)_𝐅^V̇İİ⊕ ( 1 , 1 , 0)_𝐅^V̇İİ^''⊂8_F^V̇İİ ,  
                  [  ( 3 , 1 , -1/3 )_𝐅⊕ ( 3 , 1 , -2/3 )_𝐅] ⊕[  ( 1 , 3 , +1/3 )_𝐅⊕ ( 1 , 3 , +2/3 )_𝐅]    ⊕   [ ( 3 , 3 , 0 )_𝐅⊕ ( 1 , 3 , +1/3 )_𝐅^'']  ⊕[ ( 3 , 1 , -1/3 )_𝐅^''⊕  ( 1 , 1 , 0 )_𝐅^''] ⊂28_F , 
                  [ ( 1 , 3 , +2/3 )_𝐅^'⊕ ( 1 , 1 , +1 )_𝐅^''] ⊕[  ( 3 , 1 , -2/3 )_𝐅^'⊕ ( 1 , 1 , -1 )_𝐅]    ⊕   [ ( 3 , 3 , 0 )_𝐅^'⊕  ( 1 , 3 , +1/3 )_𝐅^'⊕( 3 , 3 , +1/3 )_𝐅⊕ ( 1 , 3 , +2/3)_𝐅^'']    ⊕   [ ( 3 , 3 , 0)_𝐅^''⊕ ( 3 , 3 , -1/3)_𝐅⊕ ( 3 , 1 , -1/3)_𝐅^'''''⊕ ( 3 , 1 , -2/3)_𝐅^''']   ⊂56_F .
        
        
        We use the slashes and the back slashes to cross out massive fermions at the first and the second stages, respectively.
        From the anomaly-free conditions of [  SU(3)_c]^2 · U(1)_X_2=0, SU(3)_W
        ^2 · U(1)_X_2=0, and U(1)_X_2
        ^3=0, we find that one of the 8_F^ω and one of the 8_F^ω̇ are integrated out.
        Without loss of generality, we choose the massive anti-fundamental fermions to be ω= V and ω̇=V̇İİ at this stage.
        
        
        
        
        
        
         §.§ The third stage
        
        The third symmetry breaking stage of _331→_ SM can be achieved by Higgs fields of ( 1 , 3 ,  - 1/3 )_H ,ω⊂8_H_ , ω and ( 1 ,  3 , -1/ 3 )_H , ω̇^'⊕ ( 1 ,  3 , -1/ 3 )_H , ω̇
        ⊂28_H_ , ω̇, according to the decompositions in Eqs. (<ref>) and (<ref>), as well as their U(1)_T^'''-neutral components in Tab. <ref>.
        
        	
        	
        	
        	
        	
        	
        
        
        
        
        The Yukawa coupling between 8_F^ω and 28_F and the corresponding vectorlike fermion masses can be expressed as
        
        
        
            Y_8_F^ω28_F8_H_ ,ω + H.c. 
                   ⊃    Y_[ ( 4 , 1 , +1/4)_𝐅^ω⊗ ( 4 , 4 , 0 )_𝐅⊕  ( 1 , 4 , -1/4)_𝐅^ω⊗ ( 1 , 6 , +1/2)_𝐅^ω]⊗ ( 1 , 4 , -1/4 )_𝐇 ,ω+ H.c.    ⊃    Y_[ ( 4 , 1 , +1/4)_𝐅^ω⊗ ( 4 , 3 , +1/12 )_𝐅⊕  ( 1 , 3 , -1/3)_𝐅^ω⊗ ( 1 , 3 , +2/3)_𝐅^ω   ⊕    ( 1 , 1 , 0)_𝐅^ω^''⊗ ( 1 , 3 , +1/3)_𝐅]⊗ ( 1 , 3 , -1/3 )_𝐇 ,ω+ H.c. 
                   ⊃    Y_[ ( 3 , 1 , +1/3)_𝐅^ω⊗ ( 3 , 3 , 0)_𝐅⊕  ( 1 , 1 , 0)_𝐅^ω⊗ ( 1 , 3 , +1/3)_𝐅^''   ⊕    ( 1 , 3 , -1/3)_𝐅^ω⊗ ( 1 , 3 , +2/3)_𝐅⊕( 1 , 1 , 0)_𝐅^ω^''⊗ ( 1 , 3 , +1/3)_𝐅]⊗⟨ ( 1 , 3 , -1/3 )_𝐇 ,ω⟩ + H.c. 
            
            
                
                   ⇒   1/√(2) Y_(_L^'_R^'^c +_L^VI_R^'  c-_L^'_R^'^c   + _L^'_R^'^c  + _L^VI^''_R^c )V_3, VI   +   1/√(2) Y_( _L^'b_R^c +_L^3_R^' c-τ_L  _R^'^c  + ν_τ  L_R^'^c  +_L^3^''_R^c ) V_3,3 + H.c.  .
        
        
        Without loss of generality, we choose ω = ( 3  ,  VI ) at this stage. 
        Thus, we can identify that ( 3 , 1 , +1/3 )_𝐅^ VI ≡_R^'^c, and ( 1 , 2 , -1/2 )_𝐅^ VI ≡  ( _L^' , - _L^' ).
        The third-generational SM fermions of (τ , ν_τ , b) only form mass mixing terms with their heavy partner fermions of ( ^' , ^' , ^'), and remain massless at this stage.
        
        
        
        
        The Yukawa coupling between 8_F^ω̇ and 56_F and the corresponding vectorlike fermion masses can be expressed as
        
        
                Y_8_F^ω̇56_F28_H_ ,ω̇ + H.c. 
           ⊃    Y_[ ( 4 , 1 , +1/4)_𝐅^ω̇⊗   ( 4 , 6 , +1/4)_𝐅⊕  ( 1 , 4 , -1/4 )_𝐅^ω̇⊗ ( 1 , 4 , +3/4 )_𝐅]  ⊗  ( 1 , 6 , -1/2)_𝐇 , ω̇  + H.c. 
           ⊃    Y_[ ( 4 , 1 , +1/4)_𝐅^ω̇⊗   ( 4 , 3 , +1/12)_𝐅⊕  ( 1 , 3 , -1/3 )_𝐅^ω̇⊗ ( 1 , 3 , +2/3 )_𝐅^']  ⊗  ( 1 , 3 , -1/3)_𝐇 , ω̇^'  + H.c. 
           ⊃    Y_[ ( 3 , 1 , +1/3)_𝐅^ω̇⊗   ( 3 , 3 , 0)_𝐅^'⊕( 1 , 1 , 0)_𝐅^ω̇⊗   ( 1 , 3 , +1/3)_𝐅^'
           ⊕     ( 1 , 3 , -1/3 )_𝐅^ω̇⊗ ( 1 , 3 , +2/3 )_𝐅^']  ⊗⟨ ( 1 , 3 , -1/3)_𝐇 , ω̇^'⟩ + H.c. 
        
        
        
           ⇒   1/√(2) Y_( _L^'''_R^'''^c+ _L^V̇İİİ_R^''' c-_L^'''_R^'''^c  + _L^'''_R^'''^c )V_3,V̇İİİ^'   +   1/√(2) Y_( _L^'''s_R^c+ _L^2̇_R^''' c-μ_L _R^'''^c  + ν_μ  L_R^'''^c )  V_3,2̇^' + H.c.  , 
        
                Y_8_F^ω̇56_F28_H_ ,ω̇ + H.c. 
           ⊃    Y_[ ( 4 , 1 , +1/4)_𝐅^ω̇⊗   ( 6 , 4 , -1/4)_𝐅⊕  ( 1 , 4 , -1/4 )_𝐅^ω̇⊗ ( 4 , 6 , +1/4 )_𝐅]  ⊗  ( 4 , 4 , 0)_𝐇 , ω̇  + H.c. 
           ⊃    Y_[ ( 4 , 1 , +1/4)_𝐅^ω̇⊗   ( 6 , 3 , -1/6)_𝐅⊕  ( 1 , 3 , -1/3 )_𝐅^ω̇⊗ ( 4 , 3 , +5/12 )_𝐅   ⊕   ( 1 , 1 ,0 )_𝐅^ω̇^''⊗ ( 4 , 3 , +1/12 )_𝐅]  ⊗  ( 4 , 3 , -1/12)_𝐇 , ω̇  + H.c. 
           ⊃    Y_[ ( 3 , 1 , +1/3)_𝐅^ω̇⊗   ( 3 , 3 ,0)_𝐅^''⊕  ( 1 , 3 , -1/3 )_𝐅^ω̇⊗ ( 1 , 3 , +2/3 )_𝐅^''   ⊕   ( 1 , 1 ,0 )_𝐅^ω̇^''⊗ ( 1 , 3 , +1/3 )_𝐅^']  ⊗⟨ ( 1 , 3 , -1/3)_𝐇 , ω̇⟩ + H.c. 
        
        
        
           ⇒   1/√(2) Y_( _L^''''_R^''''^c  -_L^'''''_R^'''''^c  +_L^'''''_R^''''' c  +_L^İẊ^''_R^''' c)V_3,İẊ+ H.c.  .  
        
        Without loss of generality, we choose ω̇= ( 2̇ , V̇İİİ ) in Eq. (<ref>) and ω̇= İẊ in Eq. (<ref>) at this stage. 
        Thus, we can identify that ( 3 , 1 , +1/3 )_𝐅^V̇İİİ≡_R^'''^c, ( 1 , 2 , -1/2 )_𝐅^V̇İİİ≡  ( _L^''' , - _L^''' ),  ( 3 , 1 , +1/3 )_𝐅^İẊ≡_R^''''^c , and ( 1 , 2 , -1/2 )_𝐅^İẊ≡  ( _L^''''' , - _L^''''' ).
        
        
        
        
        There can also be mass mixing terms from the d=5 operator
        
        
        
            c_4/M_pl56_F56_F63_H28_H_,ω̇^† + H.c.    ⊃   c_4v_U/M_pl[ (  1 , 3 , +2/3 )_𝐅^'⊗  ( 4 , 1 , -3/4 )_𝐅⊕ (  4 , 3 , +1/12 )_𝐅⊗  ( 6 , 3 , -1/6 )_𝐅   ⊕   (  4 , 3 , +5/12 )_𝐅⊗  ( 6 , 1 , -1/2 )_𝐅]⊗ ( 4 , 3 , -1/12)_𝐇 , ω̇^† + H.c.    ⊃   c_4  ζ_0 [ (  1 , 3 , +2/3 )_𝐅^'⊗  ( 1 , 1 , -1)_𝐅⊕ (  3 , 3 ,0 )_𝐅^'⊗  ( 3 , 3 , -1/3 )_𝐅   ⊕   (  3 , 3 , +1/3 )_𝐅⊗  ( 3 , 1 , -2/3 )_𝐅^''']⊗ ( 1 , 3 , -1/3)_𝐇 , ω̇^'' † + H.c. 
            
                   ⇒   c_4  ζ_0[ _L  e_R^c +u_L _R^c-d_L_R^c+_L u_R^c]V_3,İẊ^*+ H.c.
        
        
        and
        
        
        
            c_4/M_pl56_F56_F63_H28_H_,ω̇^† + H.c.    ⊃   c_4v_U/M_pl[ (  4 , 1 , -3/4 )_𝐅⊗  ( 4 , 6 , +1/4 )_𝐅⊕    ( 6 , 4 , -1/4)_𝐅⊗   ( 6 , 4 , -1/4)_𝐅]⊗ ( 1 , 6 , -1/2)_𝐇 , ω̇^† + H.c.    ⊃   c_4 ζ_0 [ (  4 , 1 , -3/4 )_𝐅⊗  ( 4 , 3 , +5/12 )_𝐅]⊗ ( 1 , 3 , -1/3)_𝐇 , ω̇^† + H.c.    ⊃   c_4 ζ_0 [ (  3 , 1 , -2/3 )_𝐅^'⊗  ( 3 , 3 , +1/3 )_𝐅⊕ (  1 , 1 , -1 )_𝐅⊗  ( 1 , 3 , +2/3)_𝐅^'']⊗ ( 1 , 3 , -1/3)_𝐇 , ω̇^† + H.c. 
                   ⇒   c_4 ζ_0 [ _L c_R^c +_L μ_R^c ](V_3,2̇^'  *+V_3,V̇İİİ^'  *)+ H.c.  .
        
        
        
        The remaining massless fermions of the _ SM are listed as follows
        
        
        
            [ ( 3 , 1 , +1/3 )_𝐅^Ω⊕ ( 1 , 1 , 0 )_𝐅^Ω]  ⊕[ ( 1 , 2 , -1/2 )_𝐅^Ω⊕ ( 1 , 1 , 0 )_𝐅^Ω^'⊕ ( 1 , 1 , 0 )_𝐅^Ω^'']  ⊂8_F^Ω ,   Ω = ( 1̇ , 2̇ , 3 )  , 
                   ( 1 , 1 , 0)_𝐅^ V⊕ ... ⊕ ( 1 , 1 , 0)_𝐅^İẊ⊂8_F^Ω ,  
                   ( 1 , 1 , 0)_𝐅^ VI^'⊕  ( 1 , 1 , 0)_𝐅^V̇İİİ^'⊕ ( 1 , 1 , 0)_𝐅^İẊ^'⊂8_F^Ω^' ,  
                   ( 1 , 1 , 0)_𝐅^ IV^''⊕ ... ⊕ ( 1 , 1 , 0)_𝐅^İẊ^''⊂8_F^Ω^'' ,  
                  [  ( 3 , 1 , -1/3 )_𝐅⊕ ( 3 , 1 , -2/3 )_𝐅]  ⊕[  ( 1 , 2 , +1/2 )_𝐅⊕ ( 1 , 1 , 0 )_𝐅⊕ ( 1 , 2 , +1/2 )_𝐅^'⊕ ( 1 , 1 , +1 )_𝐅] 
               ⊕   [ ( 3 , 2 , +1/6 )_𝐅⊕ ( 3 , 1 , -1/3 )_𝐅^'⊕ ( 1 , 2 , +1/2 )_𝐅^''⊕ ( 1 , 1 , 0 )_𝐅^'⊕ ( 3 , 1 , -1/3 )_𝐅^''⊕  ( 1 , 1 , 0 )_𝐅^'']  ⊂28_F , 
                  [  ( 1 , 2 , +1/2 )_𝐅^'''⊕  ( 1 , 1 , +1 )_𝐅^'⊕ ( 1 , 1 , +1 )_𝐅^''] ⊕[ ( 3 , 1 , -2/3 )_𝐅^'⊕ ( 1 , 1 , -1 )_𝐅] 
               ⊕   [ ( 3 , 2 , +1/6 )_𝐅^'⊕ ( 3 , 1 , -1/3 )_𝐅^'''⊕ ( 1 , 2 , +1/2 )_𝐅^''''⊕  ( 1 , 1 , 0 )_𝐅^'''
               ⊕     ( 3 , 2 , +1/6 )_𝐅^''⊕ ( 3 , 1 , +2/3 )_𝐅⊕ ( 1 , 2 , +1/2)_𝐅^'''''⊕ ( 1 , 1 , +1)_𝐅^'''] 
               ⊕   [ ( 3 , 2 , +1/6 )_𝐅^'''⊕ ( 3 , 1 , -1/3)_𝐅^''''⊕ ( 3 , 2 , -1/6)_𝐅⊕ ( 3 , 1 , -2/3)_𝐅^''
               ⊕    ( 3 , 1 , -1/3)_𝐅^'''''⊕ ( 3 , 1 , -2/3)_𝐅^''']   ⊂56_F .
        
        
        The fermions that become massive at this stage are further crossed outs.
        After this stage of symmetry breaking, there are three-generational massless SM fermions together with twenty-three left-handed massless sterile neutrinos [The number of residual left-handed massless sterile neutrinos have been precisely obtained through the `t Hooft anomaly matching in Ref. <cit.>.].
        The third-generational SM fermions are from the rank-2 chiral IRAFFS of 8_F^ω⊕28_F, while the first- and second-generational SM fermions are from the rank-3 chiral IRAFFS of 8_F^ω̇⊕56_F.
        
        
        
        
        
        
        
         §.§ A summary of the vectorlike fermion masses
        
        
        
         §.§ The d=5 bi-linear fermion operators
        
        We proceed to analyze the d=5 bi-linear fermion operators in Eqs. (<ref>) along the WSW symmetry breaking pattern.
        The operator of _^ ( 3 , 2) is decomposed as
        
        
        
        
            c_3/M_pl8_F^ω̇56_F·28_H_,κ̇^†·70_H^†+ H.c. 
                   ⊃   c_3/M_pl[ (4 ,1 ,+1/4)_𝐅^ω̇⊗(4 ,6 ,+1/4)_𝐅⊕ (1 ,4 ,-1/4)_𝐅^ω̇⊗(1 ,4 ,+3/4)_𝐅]
                   ⊗   (4 ,4 ,0)_𝐇,κ̇^†⊗ (4 ,4 ,+1/2)_𝐇^†+H.c.
                   ⊃   c_3/M_pl[ (4 ,1 ,+1/4)_𝐅^ω̇⊗(4 ,3 ,+5/12)_𝐅⊕ (1 ,3 ,-1/3)_𝐅^ω̇⊗(1 ,1 ,+1)_𝐅^''⊕(1 ,1 ,0)_𝐅^ω̇^''⊗(1 ,3 ,+2/3)_𝐅^']
                   ⊗   ⟨ (4 ,1 ,+1/4)_𝐇,κ̇^†⟩⊗ (4 ,3 ,+5/12)_𝐇^†+H.c.
                   ⊃   c_3w_4,V̇İİ/√(2)M_pl[ (3 ,1 ,+1/3)_𝐅^ω̇⊗(3 ,3 ,+1/3)_𝐅⊕ (1 ,1 ,0)_𝐅^ω̇⊗(1 ,3 ,+2/3)_𝐅^''⊕ (1 ,3 ,-1/3)_𝐅^ω̇⊗(1 ,1 ,+1)_𝐅^''
                   ⊕   (1 ,1 ,0)_𝐅^ω̇^''⊗(1 ,3 ,+2/3)_𝐅^'] ⊗⟨ (1 ,3 ,+2/3)_𝐇^'''†⟩  +H.c.
            
            
                
                   ⇒   c_3/2ζ̇_̇2̇ ( _L _R^ω̇^c +_L^ω̇_R^'''^c + _L^ω̇_R ^c+_L^ω̇^''_R^'''''^c  ) v_ EW+H.c.  ,
              
              
            and
            
            
            
                      c_3/M_pl8_F^ω̇56_F·28_H_,κ̇^†·70_H^†+ H.c. 
                   ⊃   c_3/M_pl[ (4 ,1 ,+1/4)_𝐅^ω̇⊗(4 ,6 ,+1/4)_𝐅⊕ (1 ,4 ,-1/4)_𝐅^ω̇⊗(1 ,4 ,+3/4)_𝐅]
                   ⊗   (4 ,4 ,0)_𝐇,κ̇^†⊗ (4 ,4 ,+1/2)_𝐇^†+H.c.
                   ⊃   c_3/M_pl[ (4 ,1 ,+1/4)_𝐅^ω̇⊗(4 ,3 ,+1/12)_𝐅⊕ (1 ,3 ,-1/3)_𝐅^ω̇⊗(1 ,3 ,+2/3)_𝐅^']
                   ⊗   (4 ,3 ,-1/12)_𝐇,κ̇^†⊗ (4 ,3 ,+5/12)_𝐇^†+H.c.
                   ⊃   c_3/M_pl[ (3 ,1 ,+1/3)_𝐅^ω̇⊗(3 ,3 ,0)_𝐅^'⊕ (1 ,1 ,0)_𝐅^ω̇⊗(1 ,3 ,+1/3)_𝐅^'⊕(1 ,3 ,-1/3)_𝐅^ω̇⊗(1 ,3 ,+2/3)_𝐅^']
                   ⊗   ⟨ (1 ,3 ,-1/3)_𝐇,κ̇^†⟩⊗⟨ (1 ,3 ,+2/3)_𝐇^''' †⟩+H.c.
            
             
                
                   ⇒   c_3/2ζ̇_̇3̇  (  d_L _R^ω̇^c   +_L^ω̇_R^''''^c - _L^ω̇ e_R^c  +  _L^ω̇^'_R^'''^c  ) v_ EW+H.c.  ,  
                
            
            and
            
            
            
                  c_3/M_pl8_F^ω̇56_F·28_H_,κ̇^†·70_H^†+ H.c.    ⊃   c_3/M_pl[ (4 ,1 ,+1/4)_𝐅^ω̇⊗(6 ,4 ,-1/4)_𝐅⊕    (1 ,4 ,-1/4)_𝐅^ω̇⊗(4 ,6 ,+1/4)_𝐅] 
                   ⊗   (1 , 6 , -1/2)_𝐇 ,κ̇^†⊗ (4 ,4 ,+1/2)_𝐇^†+H.c.
                   ⊃   c_3/M_pl[ (4 ,1 ,+1/4)_𝐅^ω̇⊗(6 ,3 ,-1/6)_𝐅⊕    (1 ,3 ,-1/3)_𝐅^ω̇⊗(4 ,3 ,+5/12)_𝐅⊕    (1 ,1 ,0)_𝐅^ω̇^''⊗(4 ,3 ,+1/12)_𝐅] 
                   ⊗   (1 ,3 ,-1/3)_𝐇 ,κ̇^' †⊗ (4 ,3 ,+5/12)_𝐇^†+H.c.
                   ⊃   c_3/M_pl[ (3 ,1 ,+1/3)_𝐅^ω̇⊗(3 ,3 ,0)_𝐅^''⊕    (1 ,3 ,-1/3)_𝐅^ω̇⊗(1 ,3 ,+2/3)_𝐅^''⊕    (1 ,1 ,0)_𝐅^ω̇^''⊗(1 ,3 ,+1/3)_𝐅^'] 
                   ⊗   ⟨ (1 ,3 ,-1/3)_𝐇 ,κ̇^' †⟩⊗⟨(1 ,3 ,+2/3)_𝐇^''' †⟩+H.c.
            
             
             
                   ⇒   c_3/2ζ̇_3^' (   s_L _R^ω̇^c   -_L^ω̇μ_R^c  +  _L^ω̇^'_R^'''''^c + _L^ω̇^''_R^''''^c  )  v_ EW+H.c. .
        
        
        By taking the possible flavor indices of ω̇= 1̇ , 2̇ in Eqs. (<ref>) and (<ref>), one finds the following set of mass matrices of the (d , s) and (e , μ)
        
              ( _d )_2 × 2^ direct =  c_3 /2 ( ccζ̇_3     ζ̇_3  
        ζ̇_3^'   ζ̇_3^'
        )   v_ EW , 
        
               ( _e )_2 × 2^ direct = -  c_3 /2 ( ccζ̇_3    ζ̇_3^'
        ζ̇_3     ζ̇_3^'
        )  v_ EW ,
        
        which leave the down quark and electron massless.
        
        
        
        
        For the operator of _^ (4 ,1), it is decomposed as
        
        
        
            c_4/M_pl56_F56_F·28_H_,ω̇·70_H + H.c. 
                   ⊃   c_4/M_pl[ (4 ,1 ,-3/4)_𝐅⊗(4 ,6 ,+1/4)_𝐅⊕ (6 ,4 ,-1/4)_𝐅⊗ (6 ,4 ,-1/4)_𝐅] ⊗ (4 ,4 ,0)_𝐇,ω̇⊗ (4 ,4 ,+1/2)_𝐇+H.c. 
                   ⊃   c_4/M_pl   (4 ,1 ,-3/4)_𝐅⊗ (4 ,3 ,+1/12)_𝐅⊗⟨ (4 ,1 ,+1/4)_𝐇,ω̇⟩⊗ (4 ,3 ,+5/12)_𝐇+H.c.
                   ⊃    c_4 w_4,V̇İİ/√(2)M_pl[ (3 ,1 ,-2/3)_𝐅^'⊗ (3 ,3 ,0)_𝐅^'⊕ (1 ,1 ,-1)_𝐅⊗ (1 ,3 ,+1/3)_𝐅^']⊗⟨(1 ,3 ,+2/3)_𝐇^'''⟩ + H.c.
            
                 
                ⇒   c_4/2ζ̇_̇2̇  (  u_L c_R^c + _L _R^''''^c ) v_ EW + H.c.  , 
            
                     c_4/M_pl56_F56_F·28_H_,ω̇·70_H + H.c. 
                 ⊃   c_4/M_pl[ (4 ,1 ,-3/4)_𝐅⊗ (4 ,3 ,+5/12)_𝐅] ⊗ (4 ,3 ,-1/12)_𝐇,ω̇⊗ (4 ,3 ,+5/12)_𝐇+H.c.
                   ⊃   c_4/M_pl[ (3 ,1 ,-2/3)_𝐅^'⊗ (3 ,3 ,+1/3)_𝐅^'⊕ (1 ,1 ,-1)_𝐅⊗ (1 ,3 ,+2/3)_𝐅^''] ⊗⟨ (1 ,3 ,-1/3)_𝐇,ω̇⟩⊗⟨(1 ,3 ,+2/3)_𝐇^'''⟩ + H.c.
                    ⇒   c_4/2ζ̇_̇3̇ ( -_L  c_R^c - _L  _R^'''''^c  ) v_ EW + H.c.   ,
            
                   c_4/M_pl56_F56_F·28_H_,ω̇·70_H + H.c. 
                   ⊃   c_4/M_pl[ (1 ,4 ,+3/4)_𝐅⊗(4 ,1 ,-3/4)_𝐅⊕ (4 ,6 ,+1/4)_𝐅⊗(6 ,4 ,-1/4)_𝐅] ⊗ (1 ,6 ,-1/2)_𝐇,ω̇⊗ (4 ,4 ,+1/2)_𝐇+H.c.
                   ⊃   c_4/M_pl[ (1 ,3 ,+2/3)_𝐅^'⊗ (4 ,1 ,-3/4)_𝐅⊕ (4 ,3 ,+1/12)_𝐅⊗ (6 ,3 ,-1/6)_𝐅⊕ (4 ,3 ,+5/12)_𝐅⊗(6 ,1 ,-1/2)_𝐅^'']       ⊗ (1 ,3 ,-1/3)_𝐇,ω̇^'⊗ (4 ,3 ,+5/12)_𝐇+H.c.   ⊃   c_4/M_pl[ (1 ,3 ,+2/3)_𝐅^'⊗ (1 ,1 ,-1)_𝐅⊕ (3 ,3 ,0)_𝐅^'⊗ (3 ,3 ,-1/3)_𝐅⊕ (3 ,3 ,+1/3)_𝐅⊗ (3 ,1 ,-2/3)_𝐅^''']       ⊗⟨    (1 ,3 ,-1/3)_𝐇,ω̇     ^'⟩⊗⟨ (1 ,3 ,+2/3)_𝐇^'''⟩ +H.c.
            
                    ⇒   c_4 /2ζ̇_3^'  ( - _L  _R^'''^c  + u_L _R^c - _L^'''_R^c - _L u_R^c  ) v_ EW + H.c.  .
        
        
        The bi-linear fermion product of (6 ,4 ,-1/4)_𝐅⊗ (6 ,4 ,-1/4)_𝐅⊗ (4 ,4 ,0)_𝐇,ω̇⊗ (4 ,4 ,+1/2)_𝐇+H.c. was previously found to vanish due to their anti-symmetric properties <cit.>.
        
        
        
        For the operator of _^ (5 ,1), it is decomposed as
        
        
        
            c_5/M_pl28_F56_F·8_H_ , ω·70_H  + H.c. 
                   ⊃   c_5/M_pl[(6 ,1 ,-1/2)_𝐅⊗(4 ,6 ,+1/4)_𝐅⊕ (1 ,6 ,+1/2)_𝐅⊗(1 ,4 ,-3/4)_𝐅⊕(4 ,4 ,0)_𝐅⊗(6 ,4 ,-1/4)_𝐅]   ⊗   ⟨ (1 ,4 ,-1/4)_𝐇, ω⟩⊗ (4 ,4 ,+1/2)_𝐇 + H.c. 
                   ⊃    c_5W_4,IV/√(2)M_pl[(6 ,1 ,-1/2)_𝐅⊗(4 ,3 ,+1/12)_𝐅⊕ (1 ,3 ,+1/3)_𝐅⊗(4 ,1 ,-3/4)_𝐅⊕(4 ,3 ,+1/12)_𝐅⊗(6 ,1 ,-1/2)_𝐅^''   ⊕     (4 ,1 ,-1/4)_𝐅⊗ (6 ,3 ,-1/6)_𝐅] ⊗(4 ,3 ,+5/12)_𝐇 + H.c. 
                   ⊃    c_5/√(2)ζ_1 [(3 ,1 ,-2/3)_𝐅⊗(3 ,3 ,0)_𝐅^'⊕ (1 ,3 ,+1/3)_𝐅⊗(1 ,1 ,-1)_𝐅⊕(3 ,3 ,0)_𝐅⊗(3 ,1 ,-2/3)_𝐅^'''   ⊕     (3 ,1 ,-1/3)_𝐅^''⊗ (3 ,3 ,-1/3)_𝐅] ⊗⟨(1 ,3 ,+2/3)_𝐇^'''⟩ + H.c. 
                    ⇒    c_5 /2ζ_1 (  u_L t_R^c + _L _R ^c +t_L u_R^c  + _L^''_R^c ) v_ EW + H.c.   , 
            
                  c_5/M_pl28_F56_F·8_H_ , ω·70_H+H.c.
                   ⊃   c_5/M_pl[  (6 ,1 ,-1/2)_𝐅⊗ (6 ,4 ,-1/4)_𝐅⊕ (4 ,4 ,0)_𝐅⊗ (4 ,1 ,-3/4)_𝐅] ⊗ (4 ,1 ,+1/4)_𝐇 , ω⊗ (4 ,4 ,+1/2)_𝐇 + H.c. 
                   ⊃   c_5/M_pl[  (6 ,1 ,-1/2)_𝐅⊗ (6 ,3 ,-1/6)_𝐅⊕ (4 ,3 ,+1/12)_𝐅⊗ (4 ,1 ,-3/4)_𝐅] 
                   ⊗   ⟨ (4 ,1 ,+1/4)_𝐇 ,  ω⟩⊗  (4 ,3 ,+5/12)_𝐇 + H.c. 
                   ⊃   c_5w_4,V/√(2)M_pl[  (3 ,1 ,-1/3)_𝐅⊗ (3 ,3 ,-1/3)_𝐅⊕ (3 ,1 ,-2/3)_𝐅⊗ (3 ,3 ,0)_𝐅^''⊕  (3 ,3 ,0)_𝐅⊗ (3 ,1 ,-2/3)_𝐅^'
                   ⊕     (1 , 3 ,+1/3)_𝐅^''⊗ (1 ,1 ,-1)_𝐅]⊗⟨(1 ,3 ,+2/3)_𝐇^'''⟩+ H.c. 
                   ⇒   c_5/2ζ_2 ( _L _R^c + c_L  t_R^c+t_L  c_R^c + _L_R^c ) v_ EW+H.c.  , 
            
                     c_5/M_pl28_F56_F·8_H_ , ω·70_H+H.c.
                   ⊃   c_5/M_pl[(6 ,1 ,-1/2)_𝐅⊗(4 ,6 ,+1/4)_𝐅⊕ (1 ,6 ,+1/2)_𝐅⊗(1 ,4 ,-3/4)_𝐅⊕(4 ,4 ,0)_𝐅⊗(6 ,4 ,-1/4)_𝐅] 
                   ⊗   (1 ,4 ,-1/4)_𝐇 , ω⊗(4 ,4 ,+1/2)_𝐇 + H.c. 
                
               
               
                   ⊃   c_5/M_pl[(3 ,1 ,-2/3)_𝐅⊗(3 ,3 ,+1/3)_𝐅⊕ (1 ,3 ,+2/3)_𝐅⊗(1 ,1 ,-1)_𝐅⊕(3 ,3 ,0)_𝐅⊗(3 ,3 ,-1/3)_𝐅]
                   ⊗   ⟨ (1 ,3 ,-1/3)_𝐇 , ω⟩⊗⟨(1 ,3 ,+2/3)_𝐇^'''⟩+H.c.
                   ⇒   c_5/2ζ_3 ( - _L t_R^c +_L _R^'^c+ t_L  _R^c  +  _L^'_R^c ) v_ EW+H.c.  .
        
        
        
         §.§ The d=5 irreducible Higgs mixing operators
        
        We further decompose the d=5 irreducible Higgs mixing operators along the WSW symmetry breaking pattern.
        For the Yukawa coupling of 8_F^ω_128_F8_H_ ,ω_1, we find the mass terms of
        
        
        
            Y_8_F^ω_128_F8_H_ ,ω_1×d_𝒜/M_plϵ_ω_1 ω_2ω_3ω_4 8_H_ ,ω_1^†8_H_ ,ω_2^†8_H_ ,ω_3^†8_H_ ,ω_4^†70_H^†+H.c. 
                   ⊃    Y_[ (4  ,1 ,+1/4)_𝐅^ω_1⊗ ( 4 ,4 ,0)_𝐅⊕  (1 ,4 ,-1/4)_𝐅^ω_1⊗  (1 ,6 ,+1/2)_𝐅] ⊗  (1 ,4 ,-1/4)_𝐇 ,ω_1
                   ×   d_𝒜/M_pl  (1 ,4 ,-1/4)_𝐇 ,ω_1^†⊗(1 ,4 ,-1/4)_𝐇 ,ω_2^†⊗(1 ,4 ,-1/4)_𝐇 ,ω_3^†⊗ (4 ,1 ,+1/4)_𝐇 ,ω_4^†⊗ (4 ,4 ,+1/2)_𝐇^†+H.c. 
                   ⊃   Y_[ (4  ,1 ,+1/4)_𝐅^ω_1⊗ ( 4 ,3 ,+1/12)_𝐅⊕  (1 ,3 ,-1/3)_𝐅^ω_1⊗  (1 ,3 ,+2/3)_𝐅⊕(1 ,1 ,0)_𝐅^ω_1^''⊗  (1 ,3 ,+1/3)_𝐅] 
                   ⊗     (1 ,3 ,-1/3)_𝐇 ,ω_1×d_𝒜W_4 IV/√(2)M_pl  (1 ,3 ,-1/3)_𝐇 ,ω_1^†⊗(1 ,3 ,-1/3)_𝐇 ,ω_3^†⊗ (4 ,1 ,+1/4)_𝐇 ,ω_4^†⊗ (4 ,3 ,+5/12)_𝐇^†
                   +   H.c. 
                   ⊃   Y_[ (3 ,1 ,+1/3)_𝐅^ω_1⊗ (3 ,3 ,0)_𝐅⊕(1 ,1 ,0)_𝐅^ω_1⊗ (1 ,3 ,+1/3)_𝐅^''⊕  (1 ,3 ,-1/3)_𝐅^ω_1⊗  (1 ,3 ,+2/3)_𝐅
                   ⊕    (1 ,1 ,0)_𝐅^ω_1^''⊗  (1 ,3 ,+1/3)_𝐅]  (1 ,3 ,-1/3)_𝐇 ,ω_1×d_𝒜W_4 IVω_4,V/2M_pl  (1 ,3 ,-1/3)_𝐇 ,ω_1^†⊗⟨(1 ,3 ,-1/3)_𝐇 ,ω_3^†⟩
                   ⊗   ⟨(1 ,3 ,+2/3)_𝐇^''' †⟩  +H.c. 
                   ⇒    Y_   d_𝒜/ 4  W_4 , IV  w_4 , V V_3 , VI/ M_ pl  m_( 1 , 4  , - 1/4 )_H ,3^2  (  b_L b_R^c   +  τ_L τ_R^c  + _L^3 _R^''^c   - _L^ 3^'_R^'^c +_L^ 3^''_R^c  ) v_ EW  + H.c.  .
        
        
        If we consider all three possibilities of the propagator masses, each of them leads to the (b ,τ) masses as follows
        
        
        
        m_( 1 , 4  , - 1/4 )_H ,3∼(v_441)     :      m_b = m_τ =  Y_  d_𝒜/ 4 ζ_2 ζ_13 v_ EW ,
        
        m_( 1 , 4  , - 1/4 )_H ,3∼(v_431)     :     m_b = m_τ =  Y_  d_𝒜/ 4ζ_1 ζ_23 v_ EW , 
        
        m_( 1 , 4  , - 1/4 )_H ,3∼(v_331)     :      m_b = m_τ =  Y_  d_𝒜/ 4 ζ_1 /ζ_23 v_ EW .
        
        
        The last choice coincides with our previous result in Ref. <cit.>.
        
        
        
        
        The indirect Yukawa couplings from the operator in Eq. (<ref>) are expected to generate the first- and second-generational down-type quark and charged lepton masses.
        The gauge-invariant subset of ( 28_H_ ,1̇^†28_H_ ,V̇İİ) can develop the VEV of ⟨28_H_ ,1̇^†28_H_ ,V̇İİ⟩ = 1/2 w_4 ,  1̇ w_4 ,  V̇İİ∼ ( v_431^2 ) according to the VEV assignments in Eq. (<ref>).
        Similar to the indirect Yukawa couplings in Eq. (<ref>), we should look for the EWSB components from the 28_H_ , 1̇ , 2̇ here.
        For the Yukawa coupling of 8_F^ω̇_156_F28_H_ , ω̇_1, we find the mass terms of
        
        
        
            Y_8_F^ω̇_156_F28_H_ , ω̇_1 × d_ℬ/ M_ pl28_H_ , ω̇_1 ^†28_H_ , ω̇_2 ^†70_H^†( 28_H_ ,1̇^†28_H_ ,V̇İİ) + H.c. 
               ⊃    Y_[ ( 4  , 1  ,  + 1 /4  )_F^ω̇_1⊗ ( 4  , 6  ,  + 1 /4  )_F⊕ ( 1  , 4  ,  - 1 /4  )_F^ω̇_1 ⊗ ( 1  , 4  ,  + 3 /4  )_F]  ⊗ ( 1  , 6  ,  - 1 /2  )_H , ω̇_1
               ×     d_ℬ/ M_ pl ( 1  , 6  ,  - 1 /2  )_H , ω̇_1^†⊗ ( 4  , 4  ,  0 )_H , ω̇_2 ^†⊗ ( 4  , 4  ,  + 1 /2  )_H^†⊗⟨28_H_ ,1̇^†28_H_ ,V̇İİ⟩  + H.c. 
               ⊃    Y_[ ( 4  , 1  ,  + 1 /4  )_F^ω̇_1⊗ ( 4  , 3  ,  + 1 /12  )_F⊕ ( 1  , 3  ,  - 1 /3  )_F^ω̇_1 ⊗ ( 1  , 3  ,  + 2 /3  )_F^']  ⊗ ( 1  , 3  ,  - 1 /3  )_H , ω̇_1^'
               ×     d_ℬ/ M_ pl ( 1  , 3  ,  - 1 /3  )_H , ω̇_1^' †⊗ ( 4  , 3  ,  -1/12 )_H , ω̇_2 ^†⊗ ( 4  , 3  ,  + 5 /12  )_H^†⊗⟨28_H_ ,1̇^†28_H_ ,V̇İİ⟩  + H.c. 
               ⊃    Y_ d_ℬ w_4 ,  1̇ w_4 ,  V̇İİ/ 2 M_ pl m_ ( 1  , 6  ,  - 1 / 2  )_H , ω̇_1 ^2[  ( 3  , 1  ,  + 1 /3  )_F^ω̇_1⊗ ( 3  , 3  ,  0 )_F^'⊕ ( 1  , 1  , 0 )_F^ω̇_1⊗ ( 1  , 3  ,  + 1  /3  )_F^'
               ⊕    ( 1  , 3  ,  - 1 /3  )_F^ω̇_1 ⊗ ( 1  , 3  ,  +2 /3  )_F^'] ⊗⟨ ( 1  , 3  ,  - 1 /3  )_H , ω̇_2 ^†⟩⊗⟨ ( 1  , 3  ,  + 2 /3  )_H^''' †⟩ + H.c. 
               ⇒    Y_ d_ℬ/4 ζ̇_3   [    w_4 ,  1̇ w_4 ,  V̇İİ/  m_ ( 1  , 6  ,  - 1 / 2 )_H , 1̇^2    (  d_L d_R^c  +   e_L e_R^c    )  +   w_4 ,  1̇ w_4 ,  V̇İİ/  m_ ( 1  , 6  ,  - 1 / 2 )_H , 2̇^2    (  d_L s_R^c  +  μ_L e_R^c    )  ] v_ EW  + H.c.  ,
        
        
        where the SM quark/lepton components from the 8_F^ω̇_1= 1̇/8_F^ω̇_1= 2̇ correspond to the (d_R^c  , e_L) and (s_R^c  , μ_L), respectively.
        The other mass terms read
        
        
        
            Y_8_F^ω̇_156_F28_H_ , ω̇_1 × d_ℬ/ M_ pl28_H_ , ω̇_1 ^†28_H_ , ω̇_2 ^†70_H^†( 28_H_ ,1̇^†28_H_ ,V̇İİ)  + H.c. 
               ⊃    Y_[  ( 4  , 1  ,  + 1 /4  )_F^ω̇_1⊗ ( 6  , 4  ,  - 1 /4  )_F⊕ ( 1  , 4  ,  - 1 /4  )_F^ω̇_1 ⊗ ( 4  , 6  ,  + 1 /4  )_F]  ⊗ ( 4  , 4  ,  0 )_H , ω̇_1
               ×    d_ℬ/ M_ pl  ( 4  , 4  ,  0 )_H , ω̇_1^†⊗ ( 1  , 6  ,  - 1 /2  )_H , ω̇_2 ^†⊗ ( 4  , 4  ,  + 1 /2  )_H^†⊗⟨28_H_ ,1̇^†28_H_ ,V̇İİ⟩ + H.c. 
               ⊃    Y_[  ( 4  , 1  ,  + 1 /4  )_F^ω̇_1⊗ ( 6  , 3  ,  - 1 /6  )_F⊕ ( 1  , 3  ,  - 1 /3  )_F^ω̇_1 ⊗ ( 4  , 3  ,  + 5 /12  )_F⊕ ( 1  , 1 ,  0 )_F^ω̇_1^''⊗ ( 4  , 3  ,  + 1 /12  )_F]  
               ⊗    ( 4  , 3  ,  -1/12 )_H , ω̇_1× d_ℬ/  M_ pl  ( 4  , 3  ,  -1/12 )_H , ω̇_1^†⊗ ( 1  , 3  ,  -1/3 )_H , ω̇_2^' †⊗( 4  , 3  ,  +5/12 )_H , ω̇_1^†
               ⊗   ⟨28_H_ ,1̇^†28_H_ ,V̇İİ⟩+H.c.
               ⊃    Y_[  ( 3  , 1  ,  + 1 /3  )_F^ω̇_1⊗ ( 3  , 3  ,  0 )_F^''⊕  ( 1  , 3  ,  - 1 /3  )_F^ω̇_1⊗   ( 1  , 3  ,  + 2 /3  )_F^''⊕ ( 1  , 1  ,  0 )_F^ω̇_1^''⊗   ( 1  , 3  ,  + 1 /3  )_F^']
               ⊗    ( 1  , 3  ,  - 1 / 3 )_H , ω̇_1^× d_ℬw_4 ,  1̇ w_4 ,  V̇İİ/ 2 M_ pl ( 1  , 3  ,  - 1 / 3 )_H , ω̇_1^†⟨ ( 1  , 3  ,  - 1 / 3 )_H , ω̇_2^ †⟩⊗  ( 1  , 3  ,  + 2 /3  )_H^''' †  + H.c.  
               ⊃    Y_ d_ℬ w_4 ,  1̇ w_4 ,  V̇İİ V_3 , V̇İİİ^'/ 2 √(2) M_ pl m_  (4  , 4  ,  0  )_H , ω̇_1^2 [  ( 3  , 1  ,  + 1 /3  )_F^ω̇_1⊗ ( 3  , 2  ,  +1 /6  )_F^'''⊕  ( 1  , 2  ,  - 1 /2  )_F^ω̇_1⊗   ( 1  , 1  ,  + 1 )_F^'''
               ⊕     ( 1  , 1  ,  0 )_F^ω̇_1^'⊗   ( 1  , 2  ,  + 1 /2  )_F^'''''⊕  ( 1  , 1  ,  0 )_F^ω̇_1^''⊗   ( 1  , 2  ,  + 1 /2  )_F^''''] ⊗⟨ ( 1  , 2  ,  + 1 /2  )_H^''' †⟩  + H.c. 
               ⇒    Y_ d_ℬ/4 ζ̇_3^'[ w_4 ,  1̇ w_4 ,  V̇İİ/  m_ ( 4 , 4  ,  0 )_H , 1̇^2    (  s_L d_R^c  +  e_L μ_R^c  )  +  w_4 ,  1̇ w_4 ,  V̇İİ/  m_ ( 4  , 4  ,   0)_H , 2̇^2    (  s_L s_R^c  + μ_L μ_R^c  )   ] v_ EW + H.c.   .
        
        
        With the Higgs VEV assignments in (<ref>), one expects the natural propagator masses of
        
        
        
            m_ ( 4  , 4  ,  0 )_H , 1̇∼ m_ ( 1  , 6  ,  - 1 /2  )_H , 1̇∼( v_431  )  ,         m_ ( 1  , 6  ,  - 1 /2  )_H , 2̇∼ m_ ( 4  , 4  ,  0 )_H , 2̇∼( v_331  )  .
        
        
        For convenience, we parametrize the following ratios of
        
        
            Δ_ω̇≡w_4 ,  1̇ w_4 ,  V̇İİ/  m_ ( 4 , 4  , 0 )_H , ω̇^2  ,   Δ_ω̇^'≡ w_4 ,  1̇ w_4 ,  V̇İİ/  m_ ( 1  , 6  ,   - 1/ 2 )_H , ω̇^2  .
        
        
        
        
        
         §.§ The SM quark/lepton masses and the CKM mixing
        
        For all up-type quarks with Q_e=+2/3, we write down the following tree-level masses from both the renormalizable Yukawa couplings and the gravity-induced terms in the basis of ≡ (u ,c ,t)
        
        
        
        
        
        
        
        
        
        _u      =   1/√(2)( ccc  
        0       c_4  ζ̇_2 /√(2)    c_5 ζ_1 /√(2)
        
        0     0      c_5 ζ_2/ √(2)
        
           c_5 ζ_1 /√(2)     c_5 ζ_2/√(2)      Y_
        ) v_ EW≈_u^ (0) + _u^ (1 )  + _u^( 2) , 
        
        _u^ (0)   =   1/√(2)( ccc  
          0     0       0   
        
          0        0      0   
        
          0       0        Y_
        )  v_ EW , 
        
         _u^ (1)   =   1/√(2)( ccc  
         0     0       c_5 ζ_1 /√(2)
        
          0         0     0    
        
          c_5 ζ_1 /√(2)     0       0   
        ) v_ EW ,
        
        _u^ (2)   =   1/√(2)( ccc  
          0      c_4 ζ̇_2  /√(2)     0  
        
          0     0     c_5 ζ_2 /√(2)
        
           0       c_5 ζ_2 /√(2)      0  
        ) v_ EW , 
        
        where we have neglected the ∼(ζ_3 v_ EW) terms in the above expansions.
        One obvious feature is that the gauge eigenstates of up quark and the charm quark do not obtain tree-level masses through the d=5 operators with the SM Higgs doublet.
        Instead, there are only off-diagonal mass mixing terms in Eqs. (<ref>) and (<ref>).
        Accordingly, we find that
        
        
        
            det^'
                _u^ (0)_u^ (0) †
               =   1/2  Y_^2    v_ EW^2  ⇒ m_t^2 ≈ 1 /  2  Y_^2   v_ EW^2  .
        
        Here and below, we use the det^' to denote the matrix determinant that is equal to the products of all non-zero eigenvalues.
        Next, we find the charm quark mass squared of
        
        
            m_c^2    =    det^'
                ( _u^ (0) + _u^ (1)) ·( _u^ (0) † + _u^ (1) †)
            /  det^'
                _u^ (0)_u^ (0) †
            ≈   c_5^4  ζ_1^4 /8 Y_^2   v_ EW^2  .
        
        
        The up quark mass squared can be similarly obtained by
        
        
            m_u^2    =    det
                ( _u^ (0) + _u^ (1) + _u^ (2)) ·( _u^ (0) † + _u^ (1) † + _u^ (2) †)
            
                       /  det^'
                ( _u^ (0) + _u^ (1)) ·( _u^ (0) † + _u^ (1) †)
            ≈ c_4^2 ζ_2^2 ζ̇_2^2  /4 ζ_1^2   v_ EW^2  .
        
        To summarize, all SM up-type quark masses are expressed as follows
        
        
            m_u ≈ c_4 ζ_2 ζ̇_2 / 2 ζ_1  v_ EW ,   m_c ≈ c_5^2  ζ_1^2 /  2 √(2) Y_ v_ EW ,     m_t ≈ Y_/√(2) v_ EW .
        
        For all down-type quarks with Q_e=-1/3, we find the following tree-level SM mass matrix 
        
        
        
            ( _d )_3× 3≈1/4( ccc
              ( 2 c_3  +  Y_ d_ℬ  )  ζ̇_3       (  2 c_3  +  Y_ d_ℬζ_23 ^-2  ) ζ̇_3        0   
            
               ( 2 c_3 + Y_ d_ℬ  )ζ̇_3^'     ( 2 c_3  +  Y_ d_ℬζ_23 ^-2 ) ζ̇_3^'    0   
            
              0     0       Y_ d_𝒜ζ_23^-1ζ_1   
            ) v_ EW .
        
        
        For convenience, we parametrize all _331-breaking VEVs as follows
        
        
            ζ_3 = ζ̇_3^' =  ζ̇_3 /tanλ  .
        
        
        It is straightforward to find the following SM down-type quark masses of
        
        
        
             m_b    ≈    1 / 4    Y_ d_𝒜ζ_ 23 ^-1ζ_1   v_ EW ,  
        
        m_s     ≈   1 /4  ( 2c_3 + Y_ d_ℬζ_ 23 ^-2  ) ζ̇_3   v_ EW ,  
        
         m_d    ≈    c_3 ζ̇_3       v_ EW ,
        
        
        from Eq. (<ref>).
        
        
        
        
        
        For all charged leptons with Q_e=-1, their tree-level mass matrix is correlated with the down-type quark mass matrix as
        
        
        
            ( _ℓ)_ 3× 3   =   ( _d^T )_3× 3   =   1/4( ccc
                ( 2 c_3  +  Y_ d_ℬΔ_1̇^' ) ζ̇_3      ( 2 c_3 + Y_ d_ℬ  )ζ̇_3^'     0   
            
                (  2 c_3  +  Y_ d_ℬζ_23 ^-2  ) ζ̇_3       ( 2 c_3  +  Y_ d_ℬΔ_2̇   )  ζ̇_3^'    0   
            
                0     0       Y_ d_𝒜ζ_23^-1ζ_1   
            ) v_ EW .
        
        
        Thus, it is straightforward to find the tree-level mass relations of
        
        
            m_τ = m_b  ,   m_μ = m_s  ,   m_e = m_d   .
        
        
        
         §.§ The CKM matrix and the benchmark
        
        The bi-unitary transformations of
        
        
        
            _L __R^† = _^ diag ,  __^† = _L^† ( _^ diag )^2  _L   ,   = ( , )  ,
        
        
        diagonalize the un-hatted flavor eigenstates into their hatted mass eigenstates.
        To obtain the CKM matrix of the quark sector, we derive the left-handed mixing matrices of (_L  , _L) of
        
        
        
            ( cû_L 
            ĉ_L  
            t̂_L  
            )  = _L ·( c  u_L 
              c_L  
              t_L  
            )   ,   ( cd̂_L 
            ŝ_L  
            b̂_L  
            )  = _L ·( c  d_L 
              s_L  
              b_L  
            )  ,
        
        
        through their perturbative expansions in Eqs. (<ref>) and (<ref>).
        Explicitly, we find that
        
        
        
            _L = _L^ (12)·_L^ (23)·_L^ (13)·_L^ID ,
                  _L^ ID = ( ccc
            0        1      0    
            
             -1     0         0    
            
            0      0        1     
            )   ,   
            _L^ (12) = ( ccccosϵ_1      -sinϵ_1     0    
            sinϵ_1   cosϵ_1      0    
            
             0      0        1     
            )   ,   
             sinϵ_1≃m_u/m_cζ_1/ζ_2-ζ_2/ζ_1∼ (10^-2)  , 
                     _L^(13)≈( ccc
            1       0       - c_5 ζ_2/√(2)Y_
            
             0      1       0   
            c_5 ζ_2/√(2)Y_     0       1  
            )   ,   
            _L^ (23)≈( ccc
            1      0       0   
            
             0      1     c_5 ζ_1/√(2)Y_
            
             0      -c_5 ζ_1/√(2)Y_     1   
            )   ,
        
        
        and
        
        
        
            _L= ( cccsinλ   cosλ   
            
            - cosλ   sinλ   
                   1   
            )  ≈( cccλ     1-λ^2/2      
            
            - 1 + λ^2/2     λ   
                   1   
            )  .
        
        
        The CKM matrix can be approximated as the Wolfenstein parametrization
        
        
        
            V̂_ CKM|_ SU(8) ,  WSW   =   _L _L^†≈( ccc
            1-λ^2/2     λ    -c_5 ζ_2 /Y_
            
            -λ     1-λ^2/2        c_5 ζ_1 /Y_
            
              c_5  ( λζ_1 + ζ_2 ) /Y_     - c_5 ζ_1/Y_    1 
            )  ,
        
        
        which is identical to what we have obtained in the SWW sequence <cit.>.
        For completeness, we tabulate the benchmark point of the SU(8) along the WSW sequence, as well as the predicted SM quantities in Tab. <ref>.
        Three dimensionless parameters of (ζ_1 , ζ_2 , ζ_3) can be translated into three intermediate symmetry breaking scales in Eq. (<ref>) as below
        
        
                v_441≃ 1.4 × 10^17   GeV ,    v_431≃  4.8× 10^15  GeV ,    v_331≃ 4.8× 10^13  GeV .
        
        
        
        § THE WWS SYMMETRY BREAKING PATTERN
        
        
        
         §.§ The first stage
        
        The first symmetry breaking stage of _441→_431 along the WWS symmetry breaking pattern is completely identical to what we have described in Sec. <ref>.
        Thus, the remaining massless fermions can be found in Eq. (<ref>).
        
        
        
        
        
        
        
         §.§ The second stage
        
        The Yukawa coupling between 8_F^ω and 28_F and the corresponding vectorlike fermion masses can be expressed as
        
        
        
            Y_8_F^ω28_F8_H_ ,ω + H.c. 
                   ⊃   Y_[ ( 4 , 1 , +1/4)_𝐅^ω⊗ ( 4 , 4 , 0 )_𝐅⊕  ( 1 , 4 , -1/4)_𝐅^ω⊗ ( 1 , 6 ,  +1/2 )_𝐅]⊗ ( 1 , 4 , -1/4 )_𝐇 ,ω+ H.c. 
                   ⊃   Y_[ ( 4 , 1 , +1/4)_𝐅^ω⊗ ( 4 , 3 , +1/12 )_𝐅⊕  ( 1 , 3 , -1/3)_𝐅^ω⊗ ( 1 , 3 , +2/3)_𝐅
                    ⊕    ( 1 , 1 , 0)_𝐅^ω^''⊗ ( 1 , 3 , +1/3)_𝐅]⊗⟨ ( 1 , 3 , -1/3 )_𝐇 ,ω⟩ + H.c. 
            
            
            
            
                   ⇒   1/√(2)Y_( _L^'_R^'^c + _L^'_R^' c - _L^'_R^'^c   + _L^'_R^'^c + _L^V^''_R^c) w_3, V + H.c.   .
        
        
        Without loss of generality, we choose ω =  V at this stage. Thus, we can identify that ( 3 , 1 , +1/3 )_𝐅^ V ≡_R^'^c  , ( 1 , 2 , -1/2 )_𝐅^ V ≡ ( _L^' , - _L^' ), and _L^ V≡_L^'.
        
        
        
        
        The Yukawa coupling between 8_F^ω̇ and 56_F and the corresponding vectorlike fermion masses can be expressed as
        
        
        
            Y_8_F^ω̇56_F28_H_ ,ω̇ + H.c.    ⊃    Y_[ ( 4 , 1 , +1/4)_𝐅^ω̇⊗   ( 4 , 6 , +1/4)_𝐅⊕  ( 1 , 4 , -1/4 )_𝐅^ω̇⊗ ( 1 , 4 , +3/4 )_𝐅]  ⊗  ( 1 , 6 , -1/2)_𝐇 , ω̇ + H.c. 
               ⊃    Y_[  ( 4 , 1 , +1/4)_𝐅^ω̇⊗   ( 4 , 3 , +1/12)_𝐅^'⊕  ( 1 , 3 , -1/3 )_𝐅^ω̇⊗ ( 1 , 3 , +2/3 )_𝐅^']  ⊗⟨ ( 1 , 3 , -1/3)_𝐇 , ω̇^'⟩ + H.c. 
            
            
               ⇒   1/√(2) Y_(  _L^'''_R^'''^c + _L^'''_R^''' c  - _L^'''_R^'''^c +   _L^'''_R^'''^c   ) w_3 , V̇İİ
               +   1/√(2) Y_(  _L^'''d_R^c + ν_e L_R^''' c  - e_L  _R^'''^c +  _L^1̇_R^'''^c   ) w_3 , 1̇ + H.c.  .
        
        
        Without loss of generality, we choose ω̇= ( 1̇ , V̇İİ ) at this stage. Thus, we can identify that ( 3 , 1 , +1/3 )_𝐅^V̇İİ≡_R^'''^c, ( 1 , 2 , -1/2 )_𝐅^V̇İİ≡  ( _L^''' , - _L^''' ), and _L^V̇İİ≡_L^'''.
        
        
        
        
        
        
        
        
        
        
        
        
        
            
        
            
        
        
        
        
        
        After integrating out the massive fermions, the remaining massless fermions expressed in terms of the _421 irreps are the following:
        
        
        
            ( 4 , 1 , +1/4 )_𝐅^Ω⊕[  ( 1 , 2 , -1/2 )_𝐅^Ω⊕ ( 1 , 1 , 0 )_𝐅^Ω ^']  ⊕  ( 1 , 1 , 0 )_𝐅^Ω^''⊂8_F^Ω , 
                  Ω = ( ω , ω̇)  ,   ω = (3 ,  VI) ,    ω̇= (1̇ , 2̇ , V̇İİİ , İẊ ) ,
                   ( 1 , 1 , 0)_𝐅^ IV^''⊂8_F^ IV ,    ( 1 , 1 , 0)_𝐅^ V^'⊕ ( 1 , 1 , 0)_𝐅^ V^''⊂8_F^ V  ,  
                    ( 1 , 1 , 0)_𝐅^V̇İİ^'⊕ ( 1 , 1 , 0)_𝐅^V̇İİ^''⊂8_F^V̇İİ ,  
                   ( 6 , 1 , -1/2 )_𝐅⊕[    ( 1 , 2 , +1/2 )_𝐅⊕ ( 1 , 1 , 0 )_𝐅⊕( 1 , 2 , +1/2 )_𝐅⊕ ( 1 , 1 , +1 )_𝐅] 
               ⊕   [ ( 4 , 2 , +1/4 )_𝐅⊕ ( 4 , 1 , -1/4 )_𝐅^]  ⊕ ( 4  , 1 , -1/4 )_𝐅^''⊂28_F , 
                  [ ( 1 , 2 , +1/2 )_𝐅^'''⊕  ( 1 , 1 , +1 )_𝐅^'⊕  ( 1 , 1 , +1 )_𝐅^''] ⊕ ( 4 , 1 , -3/4 )_𝐅
               ⊕   [  ( 4 , 2 , +1/4 )_𝐅^'⊕ ( 4 , 1 , -1/4 )_𝐅^''⊕  ( 4 , 2 , +1/4 )_𝐅⊕   ( 4 , 1 , +3/4 )_𝐅^]  
               ⊕   [  ( 6 , 2 , 0 )_𝐅⊕   ( 6 , 1 , -1/2 )_𝐅^'⊕ ( 6 , 1 , -1/2)_𝐅^'']  ⊂56_F .
        
        
        We use the slashes and the back slashes to cross out massive fermions at the first and the second stages, respectively.
        From the anomaly-free conditions of [  SU(4)_c]^2 · U(1)_X_2=0, SU(2)_W
        ^2 · U(1)_X_2=0, and U(1)_X_2
        ^3=0, we find that one of the 8_F^ω and one of the 8_F^ω̇ are integrated out.
        Without loss of generality, we choose the massive anti-fundamental fermions to be ω= V and ω̇=V̇İİ at this stage.
        
        
        
        
        
         §.§ The third stage
        
        The third symmetry breaking stage of _ 421→_ SM can be achieved by Higgs fields of ( 4 ,   1 ,  + 1/4  )_H ,ω⊂8_H_ , ω and (   4 , 1 , +1/ 4  )_H , ω̇⊕ (  4 ,   1 ,  + 1/ 4 )_H , ω̇^'
        ⊂ (   4 , 4 , 0 )_H , ω̇⊂28_H_ , ω̇, according to the decompositions in Eqs. (<ref>) and (<ref>), as well as their U(1)_T^''' charges in Tab. <ref>.
        
        
        
        	
        The Yukawa coupling between 8_F^ω and 28_F and the corresponding vectorlike fermion masses can be expressed as
        
        
        
            Y_8_F^ω28_F8_H_ ,ω + H.c.    ⊃    Y_[ ( 4 , 1 , +1/4)_𝐅^ω⊗ ( 6 , 1 , -1/2 )_𝐅⊕  ( 1 , 4 , -1/4)_𝐅^ω⊗ ( 4 , 4 , 0)_𝐅^ω]⊗ ( 4 , 1 , +1/4 )_𝐇 ,ω+ H.c. 
                   ⊃    Y_[ ( 4 , 1 , +1/4)_𝐅^ω⊗ ( 6 , 1 , -1/2 )_𝐅⊕  ( 1 , 3 , -1/3)_𝐅^ω⊗ ( 4 , 3 , +1/12)_𝐅^ω
                   ⊕    ( 1 , 1 , 0)_𝐅^ω^''⊗ ( 4 , 1 , -1/4)_𝐅^']⊗ ( 4 , 1 , +1/4 )_𝐇 ,ω + H.c. 
                   ⊃    Y_[ ( 4 , 1 , +1/4)_𝐅^ω⊗ ( 6 , 1 , -1/2)_𝐅⊕  ( 1 , 2 , -1/2)_𝐅^ω⊗ ( 4 , 2 , +1/4)_𝐅
                   ⊕    ( 1 , 1 , 0)_𝐅^ω''⊗ ( 4 , 1 , -1/4)_𝐅^']⊗ ( 4 , 1 , +1/4 )_𝐇 ,ω+ H.c. 
                
                
                
                
                   ⇒   1/√(2) Y_( _L _R^c  + _L^''_R^''^c  -_L^''_R^''^c + _L^VI ^''_R^'' c) V_4 ,  VI
                    +   1/√(2) Y_( _L b_R^c  + τ_L _R^''^c  - ν_τ  L_R^''^c + _L^3 ^''_R^'' c) V_4 , 3 + H.c. .
        
        
        Without loss of generality, we choose ω = ( 3  ,  VI ) at this stage. 
        Thus, we can identify that ( 3 , 1 , +1/3 )_𝐅^ VI≡_R ^c ), and ( 1 , 2 , -1/2 )_𝐅^ VI ≡  ( _L^'' , - _L^'' ).
        
        
        
        
        The Yukawa coupling between 8_F^ω̇ and 56_F and the corresponding vectorlike fermion masses can be expressed as
        
        
        
            Y_8_F^ω̇56_F28_H_ ,ω̇ + H.c. 
               ⊃    Y_[ ( 4 , 1 , +1/4)_𝐅^ω̇⊗   ( 6 , 4 , -1/4)_𝐅⊕  ( 1 , 4 , -1/4 )_𝐅^ω̇⊗ ( 4 , 6 , +1/4 )_𝐅]  ⊗  ( 4 , 4 , 0)_𝐇 , ω̇  + H.c. 
               ⊃    Y_[ ( 4 , 1 , +1/4)_𝐅^ω̇⊗   ( 6 , 3 , -1/6)_𝐅⊕  ( 1 , 3 , -1/3 )_𝐅^ω̇⊗ ( 4 , 3 , +5/12 )_𝐅
               ⊕     ( 1 , 1 , 0 )_𝐅^ω̇^''⊗ ( 4 , 3 , +1/12 )_𝐅^'] ⊗ ( 4 , 3 , -1/12)_𝐇 , ω̇  + H.c. 
               ⊃    Y_[ ( 4 , 1 , +1/4)_𝐅^ω̇⊗   ( 6 , 1 , -1/2)_𝐅^'⊕  ( 1 , 2 , -1/2 )_𝐅^ω̇⊗ ( 4 , 2 , +1/4 )_𝐅
               ⊕     ( 1 , 1 , 0 )_𝐅^ω̇^''⊗ ( 4 , 1 , -1/4 )_𝐅^''] ⊗⟨ ( 4 , 1 , +1/4)_𝐇 , ω̇⟩ + H.c. 
               ⇒   1/√(2) Y_( _L^''''_R^''''^c -_L^'''''_R^'''''^c  + _L^'''''_R^'''''^c  + _L^V̇İİİ^''_R^''' c)V_4 , V̇İİİ   +   1/√(2) Y_( _L^'''' s_R^c -μ_L_R^'''''^c  + ν_μ L_R^'''''^c  + _L^2̇^''_R^''' c)V_4 , 2̇+ H.c.  , 
            
                    Y_8_F^ω̇56_F28_H_ ,ω̇ + H.c. 
               ⊃    Y_[ ( 4 , 1 , +1/4)_𝐅^ω̇⊗   ( 6 , 4 , -1/4)_𝐅⊕  ( 1 , 4 , -1/4 )_𝐅^ω̇⊗ ( 4 , 6 , +1/4 )_𝐅]  ⊗ ( 4 , 4 , 0)_𝐇 , ω̇ + H.c. 
               ⊃    Y_[ ( 4 , 1 , +1/4)_𝐅^ω̇⊗   ( 6 , 1 , -1/2)_𝐅^''⊕  ( 1 , 3 , -1/3 )_𝐅^ω̇⊗ ( 4 , 3 , +1/12 )_𝐅^']  ⊗  ( 4 , 1 , +1/4)_𝐇 , ω̇^' + H.c. 
               ⊃    Y_[ ( 4 , 1 , +1/4)_𝐅^ω̇⊗   ( 6 , 1 ,-1/2)_𝐅^''⊕  ( 1 , 2 , -1/2 )_𝐅^ω̇⊗ ( 4 , 2 , +1/4 )_𝐅^'
               ⊕    ( 1 , 1 ,0 )_𝐅^ω̇^'⊗ ( 4 , 1 , -1/4 )_𝐅^'']  ⊗⟨ ( 4 , 1 , +1/4)_𝐇 , ω̇^'⟩ + H.c. 
               ⇒   1/√(2) Y_( _L^'''''_R^'''''^c  -_L^''''_R^''''^c  +_L^''''_R^''''^c  +_L^İẊ^'_R^''' c)V_4 , İẊ^' + H.c.  .
        
        
        Without loss of generality, we choose ω̇= ( 2̇ , V̇İİİ ) and ω̇=  İẊ at this stage. 
        Thus, we can identify that ( 3 , 1 , +1/3 )_𝐅^V̇İİİ ,  İẊ≡  ( _R^''''^c  ,  _R^'''''^c), ( 1 , 2 , -1/2 )_𝐅^V̇İİİ≡  ( _L^''''' , - _L^''''' ), and ( 1 , 2 , -1/2 )_𝐅^İẊ≡  ( _L^'''' , - _L^'''' ).
        
        
        
        
        
        
        
        
        
         
        
        
         
         
         
        
         
        
        
        
        The ( 10_F , 10_F )-pair within the 56_F can obtain the vectorlike masses from the d=5 operator as follows
        
        
        
            c_4/M_pl56_F56_F⟨63_H⟩28_H_,ω̇^† + H.c. 
                   ⊃   c_4v_U/M_pl[ (  1 , 4 , +3/4 )_𝐅⊗  ( 4 , 1 , -3/4 )_𝐅⊕ (  4 , 6 , +1/4 )_𝐅⊗  ( 6 , 4 , -1/4 )_𝐅]⊗ ( 4 , 4 , 0)_𝐇 , ω̇^† + H.c. 
                   ⊃   c_4 ζ_0 [ (  1 , 1 , +1 )_𝐅^''⊗  ( 4 , 1 , -3/4 )_𝐅⊕  (  4 , 3 , +5/12 )_𝐅⊗  ( 6 , 3 , -1/6 )_𝐅]⊗ ( 4 , 1 , +1/4)_𝐇 , ω̇^' † + H.c. 
                   ⊃   c_4 ζ_0 [  (  1 , 1 , +1 )_𝐅^''⊗  ( 4 , 1 , -3/4 )_𝐅⊕  (  4 , 2 , +1/4 )_𝐅⊗  ( 6 , 2 , 0 )_𝐅
                      ⊕ (  4 , 1 , +3/4 )_𝐅⊗  ( 6 , 1 , -1/2)_𝐅^'] ⊗⟨ ( 4 , 1 , +1/4)_𝐇 , ω̇^' †⟩ + H.c. 
                   ⇒   c_4/√(2)ζ_0 ( _L _R^c + _L _R^c -_L _R^c +_L  _R^c ) V_4,İẊ^' + H.c.  .
        
        
        Similar to Eq. (<ref>), we find that the ( 10_F , 10_F )-pair obtain the vectorlike masses through the Yukawa couplings with the components of ( 4 , 4 , 0)_𝐇 , ω̇⊂28_H_ , ω̇.
        According to the VEV assignments in Eq. (<ref>), these components only contribute to the _421-breaking VEVs at the third stage.
        Correspondingly, the ( 10_F , 10_F )-pair of ( ,  ,   , ) obtain the masses of ∼(v_421).
        
        
        
        
        
        The remaining massless fermions of the _ SM are listed as follows
        
        
        
            [ ( 3 , 1 , +1/3 )_𝐅^Ω⊕ ( 1 , 1 , 0 )_𝐅^Ω]  ⊕[ ( 1 , 2 , -1/2 )_𝐅^Ω⊕ ( 1 , 1 , 0 )_𝐅^Ω^'⊕ ( 1 , 1 , 0 )_𝐅^Ω^'']  ⊂8_F^Ω ,   Ω = ( 1̇ , 2̇ , 3 )  , 
                   ( 1 , 1 , 0)_𝐅^Ω⊂8_F^Ω ,   Ω = (  VI , V̇İİİ , İẊ )  , 
                   ( 1 , 1 , 0)_𝐅^Ω^'⊂8_F^Ω ,   Ω = (  V , VI , V̇İİ ,V̇İİİ , İẊ)  , 
                   ( 1 , 1 , 0)_𝐅^Ω^''⊂8_F^Ω ,   Ω = ( IV  ,   V , VI , V̇İİ ,V̇İİİ , İẊ )  , 
                  [ ( 3 , 1 , -1/3 )_𝐅⊕ ( 3 , 1 , -2/3 )_𝐅] ⊕[     ( 1 , 2 , +1/2 )_𝐅⊕ ( 1 , 1 , 0 )_𝐅⊕( 1 , 2 , +1/2 )_𝐅^'⊕ ( 1 , 1 , +1 )_𝐅]
                  ⊕[  ( 3 , 2 , +1/6 )_𝐅⊕( 1 , 2 , +1/2 )_𝐅^''⊕ ( 3 , 1 , -1/3 )_𝐅^'⊕  ( 1 , 1 ,0 )_𝐅^']  ⊕[  ( 3  , 1 , -1/3 )_𝐅^''⊕  ( 1 , 1 , 0 )_𝐅^''] ⊂28_F , 
                  [  ( 1 , 2 , +1/2 )_𝐅^'''⊕  ( 1 , 1 , +1 )_𝐅^'⊕ ( 1 , 1 , +1 )_𝐅^''] ⊕[ ( 3 , 1 , -2/3 )_𝐅^'⊕ ( 1 , 1 , -1 )_𝐅] 
                  ⊕[ ( 3 , 2 , +1/6 )_𝐅^'⊕ ( 1 , 2 , +1/2 )_𝐅^''''⊕ ( 3 , 1 , -1/3 )_𝐅^'''⊕ ( 1 , 1 , 0 )_𝐅^'''
                  ⊕( 3 , 2 , +1/6 )_𝐅^''⊕( 1 , 2 , +1/2 )_𝐅^'''''⊕ ( 3 , 1 , +2/3 )_𝐅^⊕ ( 1 , 1 , +1 )_𝐅^'''] 
                  ⊕[  ( 3 , 2 , +1/6 )_𝐅^'''⊕( 3 , 2 , -1/6 )_𝐅⊕ ( 3 , 1 , -1/3 )_𝐅^''''⊕ ( 3 , 1 , -2/3 )_𝐅^''
                  ⊕ ( 3 , 1 , -1/3 )_𝐅^'''''⊕  ( 3 , 1 , -2/3 )_𝐅^''']  ⊂56_F .
        
        
        The fermions that become massive at this stage are further crossed outs.
        After this stage of symmetry breaking, there are three-generational massless SM fermions together with twenty-three left-handed massless sterile neutrinos.
        The third-generational SM fermions are from the rank-2 chiral IRAFFS of 8_F^ω⊕28_F, while the first- and second-generational SM fermions are from the rank-3 chiral IRAFFS of 8_F^ω̇⊕56_F.
        
        
         
         
        
         §.§ A summary of the vectorlike fermion masses
        
         
         
         
        
        
        
         §.§ The d=5 bi-linear fermion operators
        
        We proceed to analyze the d=5 bi-linear fermion operators in Eqs. (<ref>) along the WWS symmetry breaking pattern.
        The operator of _^ ( 3 , 2) is decomposed as
        
        
        
            c_3/M_pl8_F^ω̇56_F·28_H_,κ̇^†·70_H^†+ H.c. 
                   ⊃   c_3/M_pl[ (4 ,1 ,+1/4)_𝐅^ω̇⊗(4 ,6 ,+1/4)_𝐅⊕ (1 ,4 ,-1/4)_𝐅^ω̇⊗(1 ,4 ,+3/4)_𝐅]
                   ⊗   (4 ,4 ,0)_𝐇,κ̇^†⊗ (4 ,4 ,+1/2)_𝐇^†+H.c.
                   ⊃   c_3/M_pl[ (4 ,1 ,+1/4)_𝐅^ω̇⊗(4 ,3 ,+5/12)_𝐅⊕ (1 ,3 ,-1/3)_𝐅^ω̇⊗(1 ,1 ,+1)_𝐅^''⊕(1 ,1 ,0)_𝐅^ω̇^''⊗(1 ,3 ,+2/3)_𝐅^']
                   ⊗   (4 ,1 ,+1/4)_𝐇,κ̇^' †⊗ (4 ,3 ,+5/12)_𝐇^†+H.c.
                   ⊃   c_3/M_pl[ (4 ,1 ,+1/4)_𝐅^ω̇⊗(4 ,2 ,+1/4)_𝐅⊕ (1 ,2 ,-1/2)_𝐅^ω̇⊗(1 ,1 ,+1)_𝐅^''⊕(1 ,1 ,0)_𝐅^ω̇^''⊗(1 ,2 ,+1/2)_𝐅^''''']
                   ⊗   ⟨  (4 ,1 ,+1/4)_𝐇,κ̇^' †⟩⊗⟨(4 ,2 ,+1/4)_𝐇^†⟩+H.c.
                
                
                
                
                   ⇒   c_3/2ζ̇_3^' (_L  _R^ω̇^c +_L^ω̇_R^'''^c+_L^ω̇_R^c+_L^ω̇^''_R^'''''^c )v_ EW+H.c.  ,
        
        
        and
        
        
        
            c_3/M_pl8_F^ω̇56_F·28_H_,κ̇^†·70_H^†+ H.c. 
                    ⊃   c_3/M_pl[ (4 ,1 ,+1/4)_𝐅^ω̇⊗(4 ,6 ,+1/4)_𝐅⊕ (1 ,4 ,-1/4)_𝐅^ω̇⊗(1 ,4 ,+3/4)_𝐅]
                   ⊗   (4 ,4 ,0)_𝐇,κ̇^†⊗ (4 ,4 ,+1/2)_𝐇^†+H.c.
                   ⊃   c_3/M_pl[ (4 ,1 ,+1/4)_𝐅^ω̇⊗(4 ,3 ,+1/12)_𝐅^'⊕ (1 ,3 ,-1/3)_𝐅^ω̇⊗(1 ,3 ,+2/3)_𝐅^']
                   ⊗   (4 ,3 ,-1/12)_𝐇 ,κ̇^†⊗ (4 ,3 ,+5/12)_𝐇^†+H.c.
                   ⊃   c_3/M_pl[ (4 ,1 ,+1/4)_𝐅^ω̇⊗(4 ,2 ,+1/4)_𝐅^'⊕(1 ,2 ,-1/2)_𝐅^ω̇⊗(1 ,1 ,+1)_𝐅^'⊕(1 ,1 ,0)_𝐅^ω̇^'⊗(1 ,2 ,+1/2)_𝐅^''']
                   ⊗   ⟨  (4 ,1 ,-1/4)_𝐇 ,κ̇^†⟩⊗⟨(4 ,2 ,+1/4)_𝐇^†⟩+H.c.
                
                
                
                
                   ⇒   c_3/2ζ̇_̇3̇(   d_L  _R^ω̇^c  +_L^ω̇_R^''''^c-_L^ω̇e_R^c +_L^ω̇^'_R^'''^c  )  v_ EW+H.c. ,
        
        
        and
        
        
        
            c_3/M_pl8_F^ω̇56_F·28_H_,κ̇^†·70_H^†+ H.c. 
                   ⊃   c_3/M_pl[ (4 ,1 ,+1/4)_𝐅^ω̇⊗(6 ,4 ,-1/4)_𝐅⊕    (1 ,4 ,-1/4)_𝐅^ω̇⊗(4 ,6 ,+1/4)_𝐅] 
                   ⊗   (1 ,6 ,-1/2)_𝐇 ,κ̇^†⊗ (4 ,4 ,+1/2)_𝐇^†+H.c.
                   ⊃   c_3/M_pl[ (4 ,1 ,+1/4)_𝐅^ω̇⊗(6 ,3 ,-1/6)_𝐅⊕    (1 ,3 ,-1/3)_𝐅^ω̇⊗(4 ,3 ,+5/12)_𝐅⊕    (1 ,1 ,0)_𝐅^ω̇^''⊗(4 ,3 ,+1/12)_𝐅] 
                   ⊗   ⟨  (1 ,3 ,-1/3)_𝐇,κ̇^†⟩⊗ (4 ,3 ,+5/12)_𝐇^†+H.c.
                   ⊃   c_3w_3,V̇İİ/√(2)M_pl[ (4 ,1 ,+1/4)_𝐅^ω̇⊗(6 ,2 ,0 )_𝐅⊕    (1 ,2 ,-1/2)_𝐅^ω̇⊗(1 ,1 ,+1)_𝐅^'''⊕    (1 ,1 ,0)_𝐅^ω̇^'⊗(4 ,2 ,+1/4)_𝐅^'''''
                   ⊕     (1 ,1 ,0)_𝐅^ω̇^''⊗(1 ,2 ,+1/2)_𝐅^''''] ⊗⟨(4 ,2 ,+1/4)_𝐇^†⟩ +H.c.
                
                
                
                
                   ⇒   c_3/2ζ̇_̇2̇ (   s_L  _R^ω̇^2    -_L^ω̇μ_R^c+_L^ω̇^'_R^'''''^c+_L^ω̇^''_R^''''^c  )v_ EW  +H.c.  .
        
        
        By taking the possible flavor indices of ω̇= 1̇ , 2̇ in mass terms from Eqs. (<ref>) and (<ref>), one finds the following set of mass matrices of the (d , s) and (e , μ)
        
              ( _d )_2 × 2^ direct =  c_3 /2 ( ccζ̇_3    ζ̇_3   
        ζ̇_2     ζ̇_2  
        )   v_ EW , 
        
               ( _e )_2 × 2^ direct = -  c_3 /2 ( ccζ̇_3    ζ̇_2   
        ζ̇_3     ζ̇_2  
        )  v_ EW ,
        
        which leave the down quark and electron massless.
        
        
        For the operator of _^ (4 ,1), it is decomposed as
        
        
        
            c_4/M_pl56_F56_F·28_H_,ω̇·70_H + H.c. 
                   ⊃   c_4/M_pl[ (4 ,1 ,-3/4)_𝐅⊗(4 ,6 ,+1/4)_𝐅⊕ (6 ,4 ,-1/4)_𝐅⊗ (6 ,4 ,-1/4)_𝐅] ⊗ (4 ,4 ,0)_𝐇 ,ω̇⊗ (4 ,4 ,+1/2)_𝐇+H.c. 
                   ⊃   c_4/M_pl[ (4 ,1 ,-3/4)_𝐅⊗ (4 ,3 ,+1/12)_𝐅^'] ⊗  (4 ,1 ,+1/4)_𝐇,ω̇^'⊗ (4 ,3 ,+5/12)_𝐇+H.c.
                   ⊃   c_4/M_pl[ (4 ,1 ,-3/4)_𝐅⊗ (4 ,2 ,+1/4)_𝐅^'] ⊗⟨  (4 ,1 ,+1/4)_𝐇,ω̇^'⟩⊗⟨(4 ,2 ,+1/4)_𝐇⟩+ H.c. 
                
                
                
                   ⇒   c_4/2ζ̇_̇3̇^' ( u_L c_R^c + _L _R^''''^c )  v_ EW + H.c.   ,
        
        
        and
        
        
        
            c_4/M_pl56_F56_F·28_H_,ω̇·70_H + H.c. 
                
                
                   ⊃   c_4/M_pl[ (4 ,1 ,-3/4)_𝐅⊗ (4 ,3 ,+5/12)_𝐅] ⊗ (4 ,3 ,-1/12)_𝐇,ω̇⊗ (4 ,3 ,+5/12)_𝐇+H.c.
                   ⊃   c_4/M_pl[ (4 ,1 ,-1/4)_𝐅⊗ (4 ,2 ,+1/4)_𝐅] ⊗⟨ (4 ,1 ,-1/4)_𝐇,ω̇⟩⊗⟨(4 ,2 ,+1/4)_𝐇⟩+H.c.
                
                   ⇒   c_4/2ζ̇_̇3̇ ( -_L c_R^c + _L _R^'''''^c   )v_ EW + H.c.   ,
        
        
        and
        
        
        
            c_4/M_pl56_F56_F·28_H_,ω̇·70_H + H.c. 
                   ⊃   c_4/M_pl[ (1 ,4 ,+3/4)_𝐅⊗(4 ,1 ,-3/4)_𝐅⊕ (4 ,6 ,+1/4)_𝐅⊗(6 ,4 ,-1/4)_𝐅] ⊗ (1 ,6 ,-1/2)_𝐇,ω̇⊗ (4 ,4 ,+1/2)_𝐇+H.c.
                   ⊃   c_4/M_pl[ (1 ,3 ,+2/3)_𝐅^'⊗ (4 ,1 ,-3/4)_𝐅⊕ (4 ,3 ,+1/12)_𝐅^'⊗ (6 ,3 ,-1/6)_𝐅⊕ (4 ,3 ,+5/12)_𝐅⊗(6 ,1 ,-1/2)_𝐅^''] 
                   ⊗   ⟨ (1 ,3 ,-1/3)_𝐇,ω̇⟩⊗ (4 ,3 ,+5/12)_𝐇+H.c.
                   ⊃    c_4 w_3,V̇İİ/√(2)M_pl[ (1 ,2 ,+1/2)_𝐅^'''⊗ (4 ,1 ,-3/4)_𝐅⊕ (4 ,2 ,+1/4)_𝐅^'⊗ (6 ,1 ,-1/2)_𝐅^'
                   ⊕    (4 ,1 ,-1/4)_𝐅^''⊗ (6 ,2 ,0)_𝐅⊕ (4 ,2 ,+1/4)_𝐅^''⊗ (6 ,1 ,-1/2)_𝐅^''] ⊗⟨(4 ,2 ,+1/4)_𝐇⟩+H.c. 
                
                
                
                
                    ⇒   c_4 /2ζ̇_̇2̇ ( _L _R^'''^c + u_L _R^c + _L^'''_R^c - _L u_R^c  )v_ EW + H.c.   .
        
        
        	
        	
        	
        	
        
        For the operator of _^ (5 ,1), it is decomposed as
        
        
        
            c_5/M_pl28_F56_F·8_H_,ω̇·70_H+H.c.  
                   ⊃   c_5/M_pl[(6 ,1 ,-1/2)_𝐅⊗(4 ,6 ,+1/4)_𝐅⊕ (1 ,6 ,+1/2)_𝐅⊗(1 ,4 ,-3/4)_𝐅⊕(4 ,4 ,0)_𝐅⊗(6 ,4 ,-1/4)_𝐅]
                   ⊗   (1 ,4 ,-1/4)_𝐇,ω̇⊗(4 ,4 ,+1/2)_𝐇
                   ⊃   c_5W_4,IV/√(2)M_pl[(6 ,1 ,-1/2)_𝐅⊗(4 ,3 ,+1/12)_𝐅^'⊕ (1 ,3 ,+1/3)_𝐅⊗(4 ,1 ,-3/4)_𝐅⊕(4 ,3 ,+1/12)_𝐅⊗(6 ,1 ,-1/2)_𝐅^''
                   ⊕     (4 ,1 ,-1/4)_𝐅⊗ (6 ,3 ,-1/6)_𝐅] ⊗(4 ,3 ,+5/12)_𝐇
                   ⊃   c_5W_4,IV/√(2)M_pl[(6 ,1 ,-1/2)_𝐅⊗(4 ,2 ,+1/4)_𝐅^'⊕ (1 ,2 ,+1/2)_𝐅⊗(4 ,1 ,-3/4)_𝐅⊕(4 ,2 ,+1/4)_𝐅⊗(6 ,1 ,-1/2)_𝐅^''
                   ⊕     (4 ,1 ,-1/4)_𝐅⊗ (6 ,2 ,0)_𝐅] ⊗⟨(4 ,2 ,+1/4)_𝐇⟩
                
                
                
                
                    ⇒    c_5 /2ζ_1    (    u_L t_R^c + _L _R ^c +t_L  u_R^c    + _L^''_R^c )  v_ EW + H.c.   ,
        
        
        and
        
        
        
            c_5/M_pl28_F56_F·8_H_,ω̇·70_H+H.c.
                   ⊃   c_5/M_pl[  (6 ,1 ,-1/2)_𝐅⊗ (6 ,4 ,-1/4)_𝐅⊕ (4 ,4 ,0)_𝐅⊗ (4 ,1 ,-3/4)_𝐅]⊗ (4 ,1 ,+1/4)_𝐇,ω̇⊗  (4 ,4 ,+1/2)_𝐇
                   ⊃   c_5/M_pl[  (6 ,1 ,-1/2)_𝐅⊗ (6 ,3 ,-1/6)_𝐅⊕ (4 ,3 ,+1/12)_𝐅⊗ (4 ,1 ,-3/4)_𝐅]⊗ (4 ,1 ,+1/4)_𝐇,ω̇⊗  (4 ,3 ,+5/12)_𝐇
                    ⊃   c_5/M_pl[  (6 ,1 ,-1/2)_𝐅⊗ (6 ,2 ,0)_𝐅⊕ (4 ,2 ,+1/4)_𝐅⊗ (4 ,1 ,-3/4)_𝐅]⊗⟨(4 ,1 ,+1/4)_𝐇,ω̇⟩⊗⟨(4 ,2 ,+1/4)_𝐇⟩
                
                
                
                
                   ⇒   c_5/2ζ_3 (  _L _R^c +c_L t_R^c+t_L c_R^c + _L  _R^c   ) v_ EW+H.c.   ,
        
        
        and
        
        
        
            c_5/M_pl28_F56_F·8_H_,ω̇·70_H+H.c.
                   ⊃   c_5/M_pl[(6 ,1 ,-1/2)_𝐅⊗(4 ,6 ,+1/4)_𝐅⊕ (1 ,6 ,+1/2)_𝐅⊗(4 ,1 ,-3/4)_𝐅⊕(4 ,4 ,0)_𝐅⊗(6 ,4 ,-1/4)_𝐅]
                   ⊗   (1 ,4 ,-1/4)_𝐇,ω̇⊗(4 ,4 ,+1/2)_𝐇
                   ⊃   c_5/M_pl[(6 ,1 ,-1/2)_𝐅⊗(4 ,3 ,+5/12)_𝐅⊕ (1 ,3 ,+2/3)_𝐅⊗(4 ,1 ,-3/4)_𝐅⊕(4 ,3 ,+1/12)_𝐅⊗(6 ,3 ,-1/6)_𝐅]
                   ⊗   ⟨(1 ,3 ,-1/3)_𝐇,ω̇⟩⊗(4 ,3 ,+5/12)_𝐇
                   ⊃   c_5w_3,V/√(2)M_pl[(6 ,1 ,-1/2)_𝐅⊗(4 ,2 ,+1/4)_𝐅⊕ (1 ,2 ,+1/2)_𝐅^'⊗(4 ,1 ,-3/4)_𝐅⊕(4 ,2 ,+1/4)_𝐅⊗(6 ,1 ,-1/2)_𝐅^'
                   ⊕    (4 ,1 ,-1/4)_𝐅⊗(6 ,2 ,0)_𝐅]⊗⟨(4 ,2 ,+1/4)_𝐇⟩
                
                
                
                
                   ⇒   c_5/2ζ_2 ( - _L t_R^c +_L _R^'^c  +t_L   _R^c  +_L^'_R^c   ) v_ EW+H.c.  .
        
        
        
        
         §.§ The d=5 irreducible Higgs mixing operators
        
        For the Yukawa coupling of 8_F^ω_128_F8_H_ ,ω_1, we find the mass terms of
        
        
        
            Y_8_F^ω_128_F8_H_ ,ω_1×d_𝒜/M_plϵ_ω_1 ω_2ω_3ω_4 8_H_ ,ω_1^†8_H_ ,ω_2^†8_H_ ,ω_3^†8_H_ ,ω_4^†70_H^†+H.c. 
                   ⊃    Y_[ (4  ,1 ,+1/4)_𝐅^ω_1⊗ ( 4 ,4 ,0)_𝐅⊕  (1 ,4 ,-1/4)_𝐅^ω_1⊗  (1 ,6 ,+1/2)_𝐅] ⊗  (1 ,4 ,-1/4)_𝐇 ,ω_1
                   ×   d_𝒜/M_pl  (1 ,4 ,-1/4)_𝐇 ,ω_1^†⊗(1 ,4 ,-1/4)_𝐇 ,ω_2^†⊗(1 ,4 ,-1/4)_𝐇 ,ω_3^†⊗ (4 ,1 ,+1/4)_𝐇 ,ω_4^†⊗ (4 ,4 ,+1/2)_𝐇^†+H.c. 
                   ⊃   Y_[ (4  ,1 ,+1/4)_𝐅^ω_1⊗ ( 4 ,3 ,+1/12)_𝐅⊕  (1 ,3 ,-1/3)_𝐅^ω_1⊗  (1 ,3 ,+2/3)_𝐅⊕(1 ,1 ,0)_𝐅^ω_1^''⊗  (1 ,3 ,+1/3)_𝐅] 
                   ⊗     (1 ,3 ,-1/3)_𝐇 ,ω_1×d_𝒜W_4 IV/√(2)M_pl  (1 ,3 ,-1/3)_𝐇 ,ω_1^†⊗(1 ,3 ,-1/3)_𝐇 ,ω_3^†⊗ (4 ,1 ,+1/4)_𝐇 ,ω_4^†⊗ (4 ,3 ,+5/12)_𝐇^† + H.c. 
                   ⊃   Y_[ (4 ,1 ,+1/4)_𝐅^ω_1⊗ (4 ,2 ,+1/4)_𝐅⊕(1 ,2 ,-1/2)_𝐅^ω_1⊗ (1 ,1 ,+1)_𝐅^⊕  (1 ,1 ,0)_𝐅^ω_1^'⊗  (1 ,2 ,+1/2)_𝐅^'
                   ⊕     (1 ,1 ,0)_𝐅^ω_1^''⊗  (1 ,2 ,+1/2)_𝐅] ⊗ (1 ,2 ,-1/2)_𝐇 ,ω_1   ×   d_𝒜W_4 IV w_3,V/2M_pl  (1 ,2 ,-1/2)_𝐇 ,ω_1^†⊗(4 ,1 ,+1/4)_𝐇 ,ω_4^†⊗ (4 ,2 ,+1/4)_𝐇^†+H.c. 
                   ⊃   Y_[ (3 ,1 ,+1/3)_𝐅^ω_1⊗ (3 ,2 ,+1/6)_𝐅⊕ (1 ,1 ,0)_𝐅^ω_1⊗  (1 ,2 ,+1/2)_𝐅⊕(1 ,2 ,-1/2)_𝐅^ω_1⊗ (1 ,1 ,+1)_𝐅^
                      ⊕  (1 ,1 ,0)_𝐅^ω_1^'⊗  (1 ,2 ,+1/2)_𝐅^'⊕ (1 ,1 ,0)_𝐅^ω_1^''⊗  (1 ,2 ,+1/2)_𝐅] ⊗ (1 ,2 ,-1/2)_𝐇 ,ω_1      ×d_𝒜W_4 , IV  w_3 , V V_4 , VI/2√(2)M_pl  (1 ,2 ,-1/2)_𝐇 ,ω_1^†⊗⟨ (1 ,2 ,+1/2)_𝐇^'''†⟩+H.c. 
                   ⇒    Y_   d_𝒜/ 4  W_4 , IV  w_3 , V V_4 , VI/ M_ pl  m_( 1 , 4  , - 1/4 )_H ,3^2  (  b_L b_R^c   +  τ_L τ_R^c  + _L^3 _R^''^c   - _L^ 3^'_R^'^c +_L^ 3^''_R^c  ) v_ EW  + H.c.  .
        
        
        If we consider all three possibilities of the propagator masses, each of them leads to the (b ,τ) masses as follows
        
        
        
        m_( 1 , 4  , - 1/2 )_H ,3∼(v_441)     :      m_b = m_τ =  Y_  d_𝒜/ 4 ζ_2 ζ_13 v_ EW ,
        
        m_( 1 , 4 , - 1/2)_H ,3∼(v_431)     :     m_b = m_τ =  Y_  d_𝒜/ 4ζ_1 ζ_23 v_ EW , 
        
        m_( 1 , 4  , - 1/2 )_H ,3∼(v_421)     :      m_b = m_τ =  Y_  d_𝒜/ 4 ζ_1 /ζ_23 v_ EW .
        
        
        The indirect Yukawa couplings from the operator in Eq. (<ref>) are expected to generate the first- and second-generational down-type quark and charged lepton masses.
        
        The gauge-invariant subset of ( 28_H_ ,1̇^†28_H_ ,V̇İİ) can develop the VEV of ⟨28_H_ ,1̇^†28_H_ ,V̇İİ⟩ = 1/2 w_3 ,  1̇ w_3 ,  V̇İİ∼ ( v_431^2 ) according to the VEV assignments in Eq. (<ref>).
        
        Similar to the indirect Yukawa couplings in Eq. (<ref>), we should look for the EWSB components from the 28_H_ , 1̇ , 2̇ here.
        For the Yukawa coupling of 8_F^ω̇_156_F28_H_ , ω̇_1, we find the mass terms of
        
        
        
            Y_8_F^ω̇_156_F28_H_ , ω̇_1 × d_ℬ/ M_ pl28_H_ , ω̇_1 ^†28_H_ , ω̇_2 ^†70_H^†( 28_H_ ,1̇^†28_H_ ,V̇İİ) + H.c. 
               ⊃    Y_[ ( 4  , 1  ,  + 1 /4  )_F^ω̇_1⊗ ( 4  , 6  ,  + 1 /4  )_F⊕ ( 1  , 4  ,  - 1 /4  )_F^ω̇_1 ⊗ ( 1  , 4  ,  + 3 /4  )_F]  ⊗ ( 1  , 6  ,  - 1 /2  )_H , ω̇_1
               ×     d_ℬ/ M_ pl ( 1  , 6  ,  - 1 /2  )_H , ω̇_1^†⊗ ( 4  , 4  ,  0 )_H , ω̇_2 ^†⊗ ( 4  , 4  ,  + 1 /2  )_H^†⊗⟨28_H_ ,1̇^†28_H_ ,V̇İİ⟩  + H.c. 
               ⊃    Y_[ ( 4  , 1  ,  + 1 /4  )_F^ω̇_1⊗ ( 4  , 3  ,  + 1 /12  )_F⊕ ( 1  , 3  ,  - 1 /3  )_F^ω̇_1 ⊗ ( 1  , 3  ,  + 2 /3  )_F^']  ⊗ ( 1  , 3  ,  - 1 /3  )_H , ω̇_1
               ×     d_ℬ/ M_ pl ( 1  , 3  ,  - 1 /3  )_H , ω̇_1^†⊗ ( 4  , 3  ,  -1/12 )_H , ω̇_2 ^†⊗ ( 4  , 3  ,  + 5 /12  )_H^†⊗⟨28_H_ ,1̇^†28_H_ ,V̇İİ⟩  + H.c. 
               ⊃    Y_ d_ℬ w_3 ,  1̇ w_3 ,  V̇İİ/ 2 M_ pl m_ ( 1  , 6  ,  - 1 / 2  )_H , ω̇_1 ^2 [  ( 4  , 1  ,  + 1 /4  )_F^ω̇_1⊗ ( 4  , 2  ,  +1/4 )_F^'⊕ ( 1  , 2  , -1/2 )_F^ω̇_1⊗ ( 1  , 1  ,  + 1 )_F^'
               ⊕    ( 1  , 1  ,  0 )_F^ω̇_1 ^'⊗ ( 1  , 2  ,  +1 /2  )_F^'''] ⊗⟨ ( 4  , 1  ,  + 1 /4  )_H , ω̇_2 ^†⟩⊗⟨ ( 4  , 2  ,  + 1 /4  )_H^†⟩ + H.c. 
               ⇒    Y_ d_ℬ/4 ζ̇_3   [    w_3 ,  1̇  w_3 ,  V̇İİ/  m_ ( 1  , 6  ,  - 1 / 2 )_H , 1̇^2    (  d_L d_R^c  +   e_L e_R^c    )  +   w_3 ,  1̇ w_3 ,  V̇İİ/  m_ ( 1  , 6  ,  - 1 / 2 )_H , 2̇^2    (  d_L s_R^c  +  μ_L e_R^c    )  ] v_ EW  + H.c.  ,
        
        
        where the SM quark/lepton components from the 8_F^ω̇_1= 1̇/8_F^ω̇_1= 2̇ correspond to the (d_R^c  , e_L) and (s_R^c  , μ_L), respectively.
        The other mass terms read
        
        
        
            Y_8_F^ω̇_156_F28_H_ , ω̇_1 × d_ℬ/ M_ pl28_H_ , ω̇_1 ^†28_H_ , ω̇_2 ^†70_H^†( 28_H_ ,2̇^†28_H_ ,V̇İİ)  + H.c. 
               ⊃    Y_[  ( 4  , 1  ,  + 1 /4  )_F^ω̇_1⊗ ( 6  , 4  ,  - 1 /4  )_F⊕ ( 1  , 4  ,  - 1 /4  )_F^ω̇_1 ⊗ ( 4  , 6  ,  + 1 /4  )_F]  ⊗ ( 4  , 4  ,  0 )_H , ω̇_1
                  × d_ℬ/ M_ pl  ( 4  , 4  ,  0 )_H , ω̇_1^†⊗ ( 1  , 6  ,  - 1 /2  )_H , ω̇_2 ^†⊗ ( 4  , 4  ,  + 1 /2  )_H^†⊗⟨28_H_ ,2̇^†28_H_ ,V̇İİ⟩ + H.c. 
               ⊃    Y_[  ( 4  , 1  ,  + 1 /4  )_F^ω̇_1⊗ ( 6  , 3  ,  - 1 /6  )_F⊕ ( 1  , 3  ,  - 1 /3  )_F^ω̇_1 ⊗ ( 4  , 3  ,  + 5 /12  )_F⊕ ( 1  , 1 ,  0 )_F^ω̇_1^''⊗ ( 4  , 3  ,  + 1 /12  )_F]  
                  ⊗ ( 4  , 3  ,  -1/12 )_H , ω̇_1× d_ℬ/  M_ pl  ( 4  , 3  ,  -1/12 )_H , ω̇_1^†⊗ ( 1  , 3  ,  -1/3 )_H , ω̇_2^†⊗( 4  , 3  ,  +5/12 )_H , ω̇_1^†
                  ⊗⟨28_H_ ,2̇^†28_H_ ,V̇İİ⟩+H.c.
               ⊃    Y_[  ( 4  , 1  ,  + 1 /4  )_F^ω̇_1⊗ ( 6  , 2  ,  0 )_F^⊕  ( 1  , 2  ,  - 1 /2  )_F^ω̇_1⊗   ( 4  , 1  ,  + 3 /4  )_F^⊕ ( 1  , 1  ,  0 )_F^ω̇_1^'⊗   ( 4  , 2  ,  + 1 /4  )_F^
                  ⊕ ( 1  , 1  ,  0 )_F^ω̇_1^''⊗   ( 4  , 2  ,  + 1 /4  )_F^'] ⊗ ( 4  , 2  ,  - 1 / 4 )_H , ω̇_1^× d_ℬω_3 ,  2̇ω_3 ,  V̇İİ^2  / 2√(2) M_ pl ( 4  , 2  ,  - 1 / 4 )_H , ω̇_1^ †⟨ ( 4  , 2  ,  + 1 / 4 )_H^ †⟩  + H.c.  
               ⊃    Y_ d_ℬ w_3 ,  1̇ w_3 ,  V̇İİ^2  / 2 √(2) M_ pl m_  (4  , 4 ,   0  )_H , ω̇_1^2 [  ( 3  , 1  ,  + 1 /3  )_F^ω̇_1⊗ ( 3  , 2  ,  +1 /6  )_F^'''⊕  ( 1  , 2  ,  - 1 /2  )_F^ω̇_1⊗   ( 1  , 1  ,  + 1 )_F^'''
                  ⊕  ( 1  , 1  ,  0 )_F^ω̇_1^'⊗   ( 1  , 2  ,  + 1 /2  )_F^'''''⊕  ( 1  , 1  ,  0 )_F^ω̇_1^''⊗   ( 1  , 2  ,  + 1 /2  )_F^''''] ⊗⟨ ( 1  , 2  ,  + 1 /2  )_H^''' †⟩  + H.c. 
               ⇒    Y_ d_ℬ/4 ζ̇_2   [ w_3 ,  1̇ w_3 ,  V̇İİ/  m_ ( 4 , 4  ,  0 )_H , 1̇^2    (  s_L d_R^c  +  e_L μ_R^c  )  +  w_3 ,  1̇ w_3 ,  V̇İİ/  m_ ( 4  , 4  ,  0 )_H , 2̇^2    (  s_L s_R^c  + μ_L μ_R^c  )   ] v_ EW + H.c.   .
        
        
        With the Higgs VEV assignments in (<ref>) and hypothesize, we expect the following natural propagator masses of
        
        
        
            m_ ( 1  , 6  ,  - 1 /2  )_H , 1̇∼( v_431  )  ,
                
                   m_ ( 1  , 6  ,  - 1 /2  )_H , 2̇∼( v_421  )   ,
                    m_ ( 4  , 4  ,  0)_H , 1̇∼( v_431  )  ,
                
                    m_ ( 4  , 4  ,  0 )_H , 2̇∼( v_421  )  .
        
        
        For convenience, we parametrize the following ratios of
        
        
        
            Δ_ω̇≡w_4 ,  1̇ w_4 ,  V̇İİ/  m_ ( 4 , 4  ,   0 )_H , ω̇^2  ,   Δ_ω̇^'≡ w_4 ,  1̇ w_4 ,  V̇İİ/  m_ ( 1  , 6  ,   - 1/ 2 )_H , ω̇^2  .
        
        
        
        
        
        
         §.§ The SM quark/lepton masses and the CKM mixing
        
        For all up-type quarks with Q_e=+2/3, we write down the following tree-level masses from both the renormalizable Yukawa couplings and the gravity-induced terms in the basis of ≡ (u ,c ,t)
        
        
        
        
        
        
        
        
        
        _u      =   1/√(2)( ccc  
        0     c_4  ζ̇_3^'/√(2)    c_5 ζ_1 /√(2)
        
        0       0       c_5 ζ_3/√(2)
        
        c_5 ζ_1/√(2)     c_5 ζ_3/√(2)      Y_
        ) v_ EW≈_u^ (0) + _u^ (1 )  + _u^( 2) , 
        
        _u^ (0)   =   1/√(2)( ccc  
          0     0       0   
        
          0        0      0   
        
          0       0        Y_
        )  v_ EW , 
        
        _u^ (1)   =   1/√(2)( ccc  
          0       0      c_5 ζ_1/√(2)
        
           0     0     0    
        
           c_5 ζ_1 /√(2)     0     0    
        ) v_ EW , 
        
         _u^ (2)   =   1/√(2)( ccc  
         0     c_4 ζ̇_3^'/√(2)   0     
        
          0        0     c_5 ζ_3  /√(2)
        
          0      c_5 ζ_3/√(2)      0   
        ) v_ EW ,
        where we have neglected the terms of ∼(ζ_3 v_ EW) in the expansion.
        One obvious feature is that the gauge eigenstates of up quark and the charm quark do not obtain tree-level masses through the d=5 operators with the SM Higgs doublet.
        Instead, there are only off-diagonal mass mixing terms in Eqs. (<ref>) and (<ref>).
        Accordingly, we find that
        
        
        
            det^'
                _u^ (0)_u^ (0) †
               =   1/2  Y_^2    v_ EW^2  ⇒ m_t^2 ≈ 1 /  2  Y_^2   v_ EW^2  .
        
        Here and below, we use the det^' to denote the matrix determinant that is equal to the products of all non-zero eigenvalues.
        Next, we find that
        
        
        
            det^'
                ( _u^ (0) + _u^ (1)) ·( _u^ (0) † + _u^ (1) †)
               ≈   1/4  c_5^4  ζ_1^4     v_ EW^4  .
        
        
        Naturally, we expect the smaller eigenvalue above to be the charm quark mass squared of
        
        
            m_c^2    =    det^'
                ( _u^ (0) + _u^ (1)) ·( _u^ (0) † + _u^ (1) †)
            /  det^'
                _u^ (0)_u^ (0) †
            ≈ c_5^4 ζ_1^4 /8 Y_^2   v_ EW^2  .
        
        The up quark mass squared can be similarly obtained by
        
        
            m_u^2    =    det
                ( _u^ (0) + _u^ (1) + _u^ (2)) ·( _u^ (0) † + _u^ (1) † + _u^ (2) †)
            
                  /  det^'
                ( _u^ (0) + _u^ (1)) ·( _u^ (0) † + _u^ (1) †)
            ≈c_4^2 ζ_3^2 ζ̇_3^2/4 ζ_1^2   v_ EW^2  .
        
        
        To summarize, all SM up-type quark masses are expressed as follows
        
        
        
            m_u ≈ c_4 ζ_3 ζ̇_3 / 2 ζ_1  v_ EW ,   m_c ≈ c_5^2  ζ_1^2 /2 √(2) Y_ v_ EW ,     m_t ≈ Y_/√(2) v_ EW .
        
        For all down-type quarks with Q_e=-1/3, we find the following tree-level SM mass matrix 
        
        
        
            ( _d )_3× 3≈1/4( ccc
               ( 2 c_3  +  Y_ d_ℬ )  ζ̇_3     (  2 c_3  +  Y_ d_ℬζ_23 ^-2  ) ζ̇_3         0   
            
               ( 2 c_3 + Y_ d_ℬ  )ζ̇_2     ( 2 c_3  +  Y_ d_ℬζ_23 ^-2  ) ζ̇_2       0   
            
              0     0       Y_ d_𝒜ζ_23^-1ζ_1   
            
            ) v_ EW .
        
        
        This mass matrix has the similar structure as in Eq. (<ref>) from the WSW pattern, while the terms in the second row receive the _431-breaking VEVs of w_3, 1̇, V̇İİ.
        It is straightforward to find the following SM down-type quark masses of
        
        
        
             m_b    ≈    1 / 4    Y_ d_𝒜ζ_ 23 ^-1ζ_1   v_ EW ,  
        
        m_s     ≈   1 /4 ( 2c_3+  Y_ d_ℬζ_ 23 ^-2 ) ζ̇_2   v_ EW ,  
        
         m_d    ≈    c_3 ζ̇_3     v_ EW ,
        
        from Eq. (<ref>).
        
        For all charged leptons with Q_e=-1, their tree-level mass matrix is correlated with the down-type quark mass matrix as
        
        
        
            ( _ℓ)_ 3× 3   =   ( _d^T )_3× 3   ≈   1/4( ccc
               ( 2 c_3  +  Y_ d_ℬ )  ζ̇_3     ( 2 c_3 + Y_ d_ℬ  )ζ̇_2       0   
            
               (  2 c_3  +  Y_ d_ℬζ_23 ^-2  ) ζ̇_3       ( 2 c_3  +  Y_ d_ℬζ_23 ^-2  ) ζ̇_2       0   
            
              0     0       Y_ d_𝒜ζ_23^-1ζ_1   
            ) v_ EW .
        
        
        Thus, it is straightforward to find the tree-level mass relations of
        
        
        
            m_τ = m_b  ,   m_μ = m_s  ,   m_e = m_d   .
        
        
        
         §.§ The CKM matrix and the benchmark
        
        The bi-unitary transformations of
        
        
        
            _L __R^† = _^ diag ,  __^† = _L^† ( _^ diag )^2  _L   ,   = ( , )  ,
        
        
        diagonalize the un-hatted flavor eigenstates into their hatted mass eigenstates.
        To obtain the CKM matrix of the quark sector, we derive the left-handed mixing matrices of (_L  , _L) of
        
        
        
            ( cû_L 
            ĉ_L  
            t̂_L  
            )  = _L ·( c  u_L 
              c_L  
              t_L  
            )   ,   ( cd̂_L 
            ŝ_L  
            b̂_L  
            )  = _L ·( c  d_L 
              s_L  
              b_L  
            )  ,
        
        
        through their perturbative expansions in Eqs. (<ref>) and (<ref>).
        Explicitly, we find that
        
        
        
            _L = _L^ (12)·_L^ (23)·_L^ (13)·_L^ID ,
                  _L^ ID = ( ccc
            0        1      0    
            
            -1     0         0    
            
            0      0        1     
            )   ,   
            _L^ (12) = ( ccccosϵ_2      -sinϵ_2     0    
            sinϵ_2   cosϵ_2      0    
            
            0      0        1     
            )   ,   
            sinϵ_2≃m_u/m_cζ_1/ζ_3-ζ_3/ζ_1∼ (ζ_13)  , 
                  _L^(13)≈( ccc
            1       0       - c_5 ζ_3/√(2)Y_
            
            0      1       0   
            c_5 ζ_3/√(2)Y_     0       1  
            )   ,   
            _L^ (23)≈( ccc
            1      0       0   
            
            0      1     c_5 ζ_1/√(2)Y_
            
            0      -c_5 ζ_1/√(2)Y_     1   
            )   ,
        
        
        and
        
        
        
            _L 
            
            
            
            
            ≈( cccζ_ 23      1- ζ_ 23 ^2/2         0   
            
                -1+ζ_ 23 ^2/2    ζ_ 23        0   
            
             0      0       1   
            )  .
        
        
        The CKM matrix can be approximated as the Wolfenstein parametrization
        
        
        
            V̂_ CKM|_ SU(8) ,  WWS   =   _L _L^†≈( ccc
             1-ζ_ 23 ^2/2   ζ_ 23     -c_5 ζ_3/√(2)Y_
            
            - ζ_ 23      1- ζ_ 23 ^2/2     c_5 ζ_1/√(2)Y_
            c_5(ζ_ 23 ζ_1+ ζ_3  )/√(2)Y_     -c_5 ζ_1 /√(2)Y_    1 
            )   ,
        
        
        where the Cabibbo mixing parameter is interpreted as the ratio of λ = | V_us | =ζ_ 23.
        The observed mixing hierarchy of |V_cb| ≫ | V_ub| is due to the hierarchy of ζ_1 ≫ζ_3, while this was due to the hierarchy of ζ_1 ≫ζ_2 in the SWW and the WSW patterns.
        To obtain the reasonable CKM mixing parameters, one has to choose a suppressed c_3=0.01 for the reasonable (d ,e) masses, and a relatively enhanced Higgs mixing parameter of d_𝒜=0.2 for the reasonable (b , τ) masses.
        
        
        
        Based on the predicted quark masses in Eqs. (<ref>), (<ref>), and the CKM matrix in Eq. (<ref>), we suggest a set of benchmark point of the SU(8) input parameters, as well as the predicted SM quantities in Tab. <ref>.
        
        Three dimensionless parameters of (ζ_1 , ζ_2 , ζ_3) can be translated into three intermediate symmetry breaking scales in Eq. (<ref>) as below
        
        
               v_441≃ 1.4 × 10^17   GeV ,    v_431≃  4.8× 10^15  GeV ,    v_421≃ 1.1× 10^15  GeV .
        
        
        Obviously, the intermediate scale at the third symmetry breaking stage is much higher than what we have obtained in the SWW and the WSW symmetry breaking pattern <cit.>.
        
        
        
        
        
        
        
        §  THE GAUGE COUPLING EVOLUTIONS IN THE SU(8) THEORY
        
        
        
        
        
        
         §.§ Gauge coupling for the _441 fermions
        
        
          §.§.§ Covariant derivatives and gauge boson masses
        
        We express the SU(4)_W ⊗ U(1)_X_0 covariant derivatives as follows
        
        
        
        i D_μψ_4   ≡    i ∂_μψ_4  +  (  g_4W A_μ^I̅ T_ SU(4)^I̅ +  g_X_0_0 𝕀_4 X_0 μ ) ·ψ_4 ,
        
        
        for the SU(4)_W fundamental representation, and
        
        
        
        i D_μψ_4   ≡     i ∂_μψ_4 +  ( - g_4W A_μ^I̅ ( T_ SU(4)^I̅ )^T + g_X_0_0 𝕀_4 X_0 μ ) ·ψ_4 ,
        
        
        for the SU(4)_W anti-fundamental representation.
        The SU(4)_W generators of T_ SU(4)^a are normalized such that ( T_ SU(4)^I̅ T_ SU(4)^J̅)= δ^I̅J̅.
        For the SU(4)_W rank-2 anti-symmetric field of ψ_6, the covariant derivative acts as follows
        
        
        
        i D_μψ_6   ≡    i ∂_μψ_6  + g_4W A_μ^I̅ ( T_ SU(4)^I̅·ψ_6  + ψ_6· T_ SU(4)^I̅ ,T )  +  g_X_0_0  X_0 μψ_6 .
        
        
        The corresponding kinematic term for the rank-2 anti-symmetric field of ψ_6 should be Tr( ψ̅_6  i ψ_6 ).
        For the singlet of ψ_1, the covariant derivative is simply given by
        
        
        
        i D_μψ_1   ≡    (  i ∂_μ  +   g_X_0_0  X_0 μ ) ψ_1 ,
        
        The explicit form for the gauge fields of g_4W A_μ^I̅  T_ SU(4)^I̅ +g_X_1_1 𝕀_4 X_1 μ can be expressed in terms of a 4× 4 matrix below
        
        
               g_4W A^I̅_μ T_ SU(4)^I̅ + g_X_0_0 𝕀_4 X_0 μ   =    g_4W ( 
        cccc  
        0        A_μ^1 - i A_μ^2        A_μ^4 - i A_μ^5       A_μ^9 - i A_μ^10
         
        A_μ^1 +  i A_μ^2       0        A_μ^6 - i A_μ^7      A_μ^11 - i A_μ^12
         
        A_μ^4 + i A_μ^5       A_μ^6 + i A_μ^7       0      A_μ^13 - i  A_μ^14
        
        A_μ^9 + i A_μ^10    A_μ^11 + i A_μ^12     A_μ^13 + i A_μ^14     0  
        )     +    g_4W/2 diag(   A_μ^3 + 1/√(3) A_μ^8  , - A_μ^3 + 1/√(3) A_μ^8   ,  -  2 /√(3)  A_μ^8   ,  0 )    +    g_4W/2 √(6) diag(  
            A_μ^15  + 12 t_ϑ_G _0 X_0 μ
        𝕀_3× 3 ,   - 3  A_μ^15 +  12 t_ϑ_G _0 X_0 μ)  ,
        
        
        where we have defined an SU(4)_W mixing angle of
        
        
        
        t_ϑ_G   ≡   tanϑ_G = g_X_1/√(6) g_4W  .
        
        
        For the flavor-conserving neutral gauge bosons of (A_μ^15 , X_1 μ ), the mass squared matrix reads
        
        
              3 /16   g_4W^2 v_441^2  (   A^15_μ , X_0 μ  )  ·( 
        cc  
        1        - t_ϑ_G
         
        - t_ϑ_G    t_ϑ_G^2   
        )  ·( c  A^15 μ
         X_0^μ
        ) .
        
        
        Obviously it contains a zero eigenvalue which corresponds to the massless gauge boson of X_2 μ after the SU(4)_W symmetry breaking.
        The gauge eigenstates of (  A^15_μ , X_1 μ) are diagonalized as follows
        
        
        ( c  Z^''_μ
         X_1 μ
        )    =   ( cc  
        c_ϑ_G      - s_ϑ_G
         
        s_ϑ_G     c_ϑ_G
        ) ·( c  A^15_μ
         X_0 μ
        ) .
        
        
        The SU(4)_W ⊗ U(1)_X_1 gauge couplings of (α_4W , α_X_1 ) match with the SU(3 )_W ⊗ U(1)_X_2 gauge couplings as follows
        
        
              α_3W ^-1 (v_441 ) =  α_4W ^-1 (v_441 )   , α_X_1^-1 (v_441 )  = 1/6α_4W ^-1 (v_441 )  +  α_X_0^-1 (v_441 )   ,       1/6α_4W ^-1 = α_X_1^-1 s_ϑ_G^2  , α_X_0^-1 = α_X_1^-1 c_ϑ_G^2   .
        
        
        
        
        The neutral currents are expressed as follows in the V-A basis
        
        
        _ SU(4)_W^ NC , F   =   g_X_1/ s_ϑ_G  c_ϑ_G ( g_f^V ''fγ^μ f + g_f^A ''fγ^μγ_5 f ) Z_μ^'' ,
        
        
        and we tabulate the vectorial and axial couplings of ( g_f^V '' , g_f^A '' ) in Tab. <ref>. 
        The flavor non-universality of the neutral currents mediated by Z_μ^'' among three generational SM quarks and leptons are displayed explicitly. 
        
        
        
        
        The gauge coupling evolutions rely on the RGEs of the SU(8) theory.
        The two-loop RGE of a gauge coupling of α_Υ in the MS scheme is given by <cit.>
        
               d α^-1_Υ ( μ) / dlogμ  = b^ (1) _Υ/2π + ∑_Υ^'b^ (2 ) _ΥΥ^'/8π^2α^-1_Υ ( μ )  ,
        
        
        where the one-loop and two-loop β coefficients for the non-Abelian groups are
        
        
        
            b^1_Υ=   -11/3C_2_Υ+2/3∑_FT^F_Υ+1/3∑_ST^S_Υ , 
            
            b^2_ΥΥ^'=   -34/3C_2_Υ^2α_Υ-∑_F
                2∑_Υ^'C_2^F_Υ^'α_Υ^'+10/3C_2_Υα_Υ
            T^F_Υ
               -∑_S
                4∑_Υ^' C_2^S_Υ^'α_Υ^'+2/3C_2_Υα_Υ
            T^S_Υ .
        
        
        Here, C_2( _Υ ) is the quadratic Casimir of the gauge group _Υ, T(  ^F_Υ ) and T(  ^S_Υ ) are trace invariants of the chiral fermions in the irrep of ^F_Υ and complex scalars in the irrep of ^S_Υ.
        For the U(1) Abelian groups with charges denoted as ^F/S, the one-loop β coefficients become
        
        
        
        b^1_Υ=2/3∑_F^F_Υ^2+1/3∑_S^S_Υ^2 .
        
        
        Below, we first list the minimal set of massless Higgs fields according to the survival hypothesis <cit.> and the massless fermions between different symmetry breaking stages according to the minimal Higgs VEVs in Eqs. (<ref>) and (<ref>).
        The two-loop RGEs within the minimal SU(8) setup are therefore derived by using the PyR@TE <cit.>.
        
        
        
        
        
        
        
         §.§ RGEs in the WSW symmetry breaking pattern
        
        Between the v_441≤μ≤ v_U, almost all massless Higgs fields are in the first lines of Eqs. (<ref>), (<ref>), and (<ref>), which are
        
        
               ( 4 , 1 , +1/4)_𝐇 , ω⊕
        ( 1 , 4 , -1/4)_𝐇 , ω⊂8_H_, ω ,        ( 6 , 1 , +)_𝐇 ,ω̇⊕
        (1 ,6 ,-)_𝐇 ,ω̇⊕
        (4 ,4 ,0)_𝐇 ,ω̇⊂28_H_,ω̇ ,       (4 ,4 ,)_𝐇⊕
        (4 ,4 ,)_𝐇⊕
        (6 ,6 ,0)_𝐇⊂70_H .
        
        
        All SU(8) fermions in Eq. (<ref>) remain massless after the decomposition into the _441 irreps.
        Correspondingly, we have the _441β coefficients of
        
        
               (b^(1)_ SU(4)_s ,b^(1)_ SU(4)_W ,b^(1)_ U(1)_X_0)=(+13/3 ,+13/3 ,+55/3) , 
               b^(2)__441=
        [ 2299/6  405/2     12;  405/2 2299/6     12;    180    180     31 ] .
        
        
        When the RGEs evolve down to v_441, the gauge couplings should match according to the following relations
        
        
              α_3W ^-1(v_441 ) = α_4W ^-1 (v_441 )  , α_X_1^-1 (v_441 ) = 1/6α_4W ^-1(v_441 ) +  α_X_0^-1 (v_441 )  .
        
        
        
        Between the v_431≤μ≤ v_441, the massless Higgs fields are
        
        
               ( 4 , 1 , +1/4)_𝐇 , V⊂8_H_, V ,        ( 1 , 3 , -1/3)_𝐇 , 3 ,VI⊂ ( 1 , 4 , -1/4)_𝐇 , 3 ,VI⊂8_H_, 3 ,VI ,       (4 ,3 ,-1/12)_𝐇 ,İẊ⊂ (4 ,4 ,    0        )_𝐇 ,İẊ⊂28_H_,İẊ ,       (4 ,1 ,+1/4)_𝐇 ,1̇ ,V̇İİ⊂ (4 ,4 ,   0 )_𝐇 ,1̇ ,V̇İİ⊂28_H_,1̇ ,V̇İİ ,        (1 ,3 ,-1/3)^'_𝐇 ,2̇ ,V̇İİİ⊂ (1 ,6 ,-1/2)_𝐇 ,2̇ ,V̇İİİ⊂28_H_,2̇ ,V̇İİİ ,        (4 ,3 ,+5/12)_𝐇⊂ (4 ,4 ,+1/ 2)_𝐇⊂70_H .
        
        
        The massless _431 fermions are listed at Eq. (<ref>).
        Correspondingly, we have the _431β coefficients of
        
        
               (b^(1)_ SU(4)_s ,b^(1)_ SU(3)_W ,b^(1)_ U(1)_X_1)=(-23/6 ,+1/3 ,+443/36) ,        b^(2)__431=
        [    659/6       36     17/4;    135/2    358/3   181/36;    255/4    362/9 2641/216 ] .
        
        
        When the RGEs evolve down to v_431, the gauge couplings should match according to the following relations
        
        
              α_3c ^-1(v_431 ) = α_4s ^-1 (v_431 )  , α_X_2^-1 (v_431 ) = 1/6α_4s ^-1(v_431 ) +  α_X_1^-1 (v_431 )  .
        
        
        
        Between the v_331≤μ≤ v_431, the massless Higgs fields are
        
        
              ( 1 , 3 , -1/3)_𝐇 , 3 ,VI⊂  ( 1 , 4 , -1/4)_𝐇 , 3 ,VI⊂8_H_, 3 ,VI ,       (1 ,3 ,-1/3)_𝐇 ,İẊ⊂(4 ,3 ,-1/12)_𝐇 ,İẊ⊂ (4 ,4 ,    0        )_𝐇 ,İẊ⊂28_H_,İẊ ,        (1 ,3 ,-1/3)^'_𝐇 ,2̇ ,V̇İİİ⊂ (1 ,6 ,-1/2)_𝐇 ,2̇ ,V̇İİİ⊂28_H_,2̇ ,V̇İİİ ,        (1 ,3 ,+2/3)_𝐇^'''⊂ (4 ,3 ,+5/12)_𝐇⊂ (4 ,4 ,+1/ 2)_𝐇⊂70_H . 
        
        
        The massless _331 fermions are listed at Eq. (<ref>).
        Correspondingly, we have the _331β coefficients of
        
        
              (b^(1)_ SU(3)_c ,b^(1)_ SU(3)_W ,b^(1)_ U(1)_X_2)=(-5 ,-4 ,+9) ,      b^(2)__331=
        [    12    12     2;    12    34     4;    16    32 100/9 ] .
        
        
        When the RGEs evolve down to v_331, the gauge couplings should match according to the following relations
        
        
              α_2W ^-1 (v_331 ) =  α_3W ^-1 (v_331 )  ,  α_Y^-1 (v_331 ) = 1/3α_3W ^-1 (v_331 ) +  α_X_2^-1 ( v_331 )  .
        
        
        
        Between the v_ EW≤μ≤ v_331, the massless Higgs fields are
        
        
        
        (1 ,2 ,+1/2)^'''_𝐇⊂ (1 ,3 ,+2/3)_𝐇^'''⊂ (4 ,3 ,+5/12)_𝐇⊂ (4 ,4 ,+1/ 2)_𝐇⊂70_H  .
        
        
        The massless _ SM fermions are listed at Eq. (<ref>).
        Correspondingly, we have the _ SMβ coefficients of
        
        
               (b^(1)_ SU(3)_c ,b^(1)_ SU(2)_W ,b^(1)_ U(1)_Y)=(-7 ,-19/6 ,+41/6) ,        b^(2)__SM=
        [    -26    9/2   11/6;     12   35/6    3/2;   44/3    9/2 199/18 ] .
        
        
        
        With the one- and two-loop β coefficients, we plot the RGEs of the minimal SU(8) setup according to the WSW symmetry breaking pattern in Fig. <ref>.
        Three intermediate symmetry breaking scales follow from the benchmark point in Eq. (<ref>).
        Obviously, the evolutions of three gauge couplings cannot achieve the unification.
        
        
        
        
        
        
         §.§ RGEs in the WWS symmetry breaking pattern
        
        Between the v_441≤μ≤ v_U, the _441β coefficients are identical to Eq. (<ref>), while the matching conditions at v_441 are identical to Eq. (<ref>).
        
        
        
        
        
        
        
        
        Between the v_431≤μ≤ v_441, the massless Higgs fields are
        
        
               ( 4 , 1 , +1/4)_𝐇 , 3 , VI⊂8_H_, 3 ,VI ,       ( 1 , 3 , -1/3)_𝐇 ,V⊂ ( 1 , 4 , -1/4)_𝐇 ,V⊂8_H_, V ,      	(4 ,3 ,-1/12)_𝐇 ,İẊ⊂ (4 ,4 ,    0        )_𝐇 ,İẊ⊂28_H_,İẊ ,       (4 ,1 ,+1/4)^'_𝐇 ,2̇ ,V̇İİİ⊂ (4 ,4 ,   0 )_𝐇 ,2̇ ,V̇İİİ⊂28_H_,1̇ ,V̇İİ ,      	(1 ,3 ,-1/3)^'_𝐇 ,1̇ ,V̇İİ⊂ (1 ,6 ,-1/2)_𝐇 ,1̇ ,V̇İİ⊂28_H_,2̇ ,V̇İİİ  ,       (4 ,3 ,+5/12)_𝐇⊂ (4 ,4 ,+1/ 2)_𝐇⊂70_H .
        
        
        The massless _431 fermions are listed at Eq. (<ref>).
        Correspondingly, we have the _431β coefficients of
        
        
               (b^(1)_ SU(4)_s ,b^(1)_ SU(3)_W ,b^(1)_ U(1)_X_1)=(-10/3 ,+5/6 ,+110/9) ,       b^(2)__431=
        [  1501/12       44   103/24;    165/2    391/3   175/36;    515/8    350/9 5219/432 ] .
        
        
        When the RGEs evolve down to v_431, the gauge couplings should match according to the following relations
        
        
              α_2W ^-1 (v_431 ) =  α_3W ^-1 (v_431 )  ,  α_X_2^-1 (v_431 ) = 1/3α_3W ^-1 (v_431 ) +  α_X_1^-1 ( v_431 )  .
        
        
        
        Between the v_421≤μ≤ v_431, the massless Higgs fields are
        
        
              ( 4 , 1 , +1/4)_𝐇 , 3 ,VI⊂8_H_, 3 ,VI ,        (4 ,1 ,+1/4)_𝐇 ,İẊ⊂ (4 ,3 ,-1/12)_𝐇 ,İẊ⊂ (4 ,4 ,    0        )_𝐇 ,İẊ⊂28_H_,İẊ ,      (4 ,1 ,+1/4)^'_𝐇 ,2̇ ,V̇İİİ⊂ (4 ,4 ,   0 )_𝐇 ,2̇ ,V̇İİİ⊂28_H_,1̇ ,V̇İİ ,       (4 ,2 ,+1/ 4)_𝐇⊂ (4 ,3 ,+5/12)_𝐇⊂ (4 ,4 ,+1/ 2)_𝐇⊂70_H .
        
        
        The massless _421 fermions are listed at Eq. (<ref>).
        Correspondingly, we have the _421β coefficients of
        
        
              (b^(1)_ SU(4)_s ,b^(1)_ SU(2)_W ,b^(1)_ U(1)_X_2)=(-11/2 ,+4/3 ,+151/12) ,       b^(2)__421=
        [ 131/2  21/2  17/4; 105/2 184/3  11/4; 255/4  33/4 125/8 ] .
        
        
        When the RGEs evolve down to v_421, the gauge couplings should match according to the following relations
        
        
              α_3c ^-1(v_421 ) = α_4s ^-1 (v_421 )  , α_Y^-1 (v_421 ) = 1/6α_4s ^-1(v_421 ) +  α_X_2^-1 (v_421 )  .
        
        
        Between the v_ EW≤μ≤ v_421, the massless Higgs fields are
        
        
        
        (1 ,2 ,+1/2)^'''_𝐇⊂ (4 ,2 ,+1/ 4)_𝐇⊂ (4 ,3 ,+5/12)_𝐇⊂ (4 ,4 ,+1/ 2)_𝐇⊂70_H .
        
        
        The massless _ SM fermions are listed at Eq. (<ref>).
        Correspondingly, we have the _ SMβ coefficients of
        
        
              (b^(1)_ SU(3)_c ,b^(1)_ SU(2)_W ,b^(1)_ U(1)_Y)=(-7 ,-19/6 ,+41/6) ,       b^(2)__SM=
        [    -26    9/2   11/6;     12   35/6    3/2;   44/3    9/2 199/18 ] .
        
        
        With the one- and two-loop β coefficients, we plot the RGEs of the minimal SU(8) setup according to the WWS symmetry breaking pattern in Fig. <ref>.
        Three intermediate symmetry breaking scales follow from the benchmark point in Eq. (<ref>).
        Neither the evolutions of three gauge couplings can achieve the unification along the WWS sequence.
        
        
        
        
        
        
        
        § SUMMARY AND OUTLOOK
        
        The WSW and the WWS symmetry breaking patterns following the maximally broken SU(8) theory have been analyzed in both the fermion spectra and the gauge coupling evolutions.
        Throughout our analyses of the both patterns, we compare our results from what we have obtained previously in Refs. <cit.>.
        
        
        
          *  Given the SU(8) chiral fermions in Eq. (<ref>), one finds six vectorlike (5_F , 5_F)-pairs and one (10_F , 10_F)-pair in addition to three-generational SM fermions through Georgi's decomposition rule <cit.>.
        Both symmetry breaking patterns are found to give arise to massive vectorlike fermions in the spectrum consist to what we have found along the SWW pattern <cit.>.
        
        
        
          *  With the central motivation to address the SM quark/lepton masses through the non-universal Yukawa couplings to the unique SM Higgs boson in the spectrum, the SM fermion flavor identifications are elaborated along both the WSW and the SWW pattern.
        In particular, we found that the flavor identifications between the first- and second-generational SM fermions are different from those in the SWW pattern, as one can compare the SM flavors in Tabs. <ref> and <ref> versus the results in Tab. <ref>.
        The flavor identifications are determined by requiring the similar up-type quark mass matrices in all three symmetry breaking patterns, as one can find in Eqs. (<ref>), (<ref>), and (<ref>).
        These flavor identifications, together with the down-type quarks, lead to both reasonable SM quark mass hierarchies as well as the observed CKM mixing pattern.
        
        
        
          *  There are also observed differences in the SM quark/lepton mass spectra between the (SWW, WSW) and the WWS patterns.
        The up-type quark mass matrix along the WWS pattern contains the dimensionless parameters of (ζ_1 , ζ_3 , ζ̇_3^') in Eq. (<ref>), rather than dimensionless parameters of (ζ_1 , ζ_2 , ζ̇_2).
        For the down-type quark and charged lepton sectors along the WWS pattern, the mass matrices contain the dimensionless parameters of (ζ̇_2 , ζ̇_3).
        Correspondingly, the Cabibbo angle is then interpreted as the ratio between two different symmetry breaking scales such that λ=|V_us|=ζ_23.
        In the (SWW, WSW) patterns, the Cabibbo angle is interpreted as the relative ratio between two _331-breaking VEVs in Eq. (<ref>).
        These differences originate from the Higgs VEVs at the third symmetry breaking stage in the rank-3 chiral IRAFFS sector since (i) along the (SWW, WSW) patterns, the Higgs VEVs at the third symmetry breaking stage come from two different Higgs components of ( 1 , 3 , -1/3 )_𝐇 , ω̇^'⊂ ( 1 , 6 , -1/2 )_𝐇 , ω̇ and ( 1 , 3 , -1/3 )_𝐇 , ω̇⊂ ( 4 , 4 , 0 )_𝐇 , ω̇ in Eq. (<ref>), and (ii) along the WWS pattern, the Higgs VEVs at the third symmetry breaking stage come from one single Higgs component of ( 4 , 1 , +1/4 )_𝐇 , ω̇⊕ ( 4 , 1 , +1/4 )_𝐇 , ω̇^'⊂ ( 4 , 4 , 0 )_𝐇 , ω̇ in Eq. (<ref>). 
        Another consequence of the above differences can be found in the suggested benchmark points in Tabs. <ref> and <ref>.
        
        
        
        
          *  The gauge coupling evolutions in both patterns are estimated according to the two-loop RGEs based on the suggested intermediate symmetry breaking scales in Eqs. (<ref>) and (<ref>).
        Their behaviors are largely matching with what we have found in the SWW sequence <cit.>, where large discrepancies of | α_X_0^-1 - ( α_4S^-1 + α_ 4W^-1 )/2 | ∼ 20 above μ∼ 10^17  GeV can be observed in both Figs. <ref> and <ref>.
        Within the field theory context, such large discrepancy could not be reconciled with the one-loop threshold effects <cit.>.
        Altogether, the gauge coupling unification should better be interpreted in the context of the Kač-Moody Lie algebra such that
        
        
               k_i g_i^2 = g_ string^2 =  8 π G_N /α^' ,
        
        
        where k_i represent the Kač-Moody levels of each gauge couplings, and α^' represents the Regge slope.
        To achieve the string unification, it is necessary to consider the constraints of the unitarity and the conformal invariance to the SU(8) algebra.
        
        
        
        
        
        
        § ACKNOWLEDGEMENTS
        
        
        
        
        We would like to thank Tianjun Li, Kaiwen Sun, Yuan Sun, Yinan Wang, Zhong-Zhi Xianyu, and Wenbin Yan for very enlightening discussions and communications. 
        N.C. would like to thank Nanjing University, Central China Normal University, and South China Normal University for hospitality when preparing this work.
        N.C. is partially supported by the National Natural Science Foundation of China (under Grants No. 12035008 and No. 12275140) and Nankai University.
        
        
        
        
        
        
        
        §  DECOMPOSITIONS OF THE SU(8) HIGGS 
        
        
         §.§  Type-WSW
        
        
         §.§  Type-WWS
        
        This section include Higgs part of Type-WWS model.
        
        
        
         
        
        
        
        
        
        
        § MAIN RESULTS IN THE SU(8) SWW SYMMETRY BREAKING PATTERN
        
        
        
        
        
        
        
        By following the symmetry breaking pattern in Eq. (<ref>) and the U(1) charges defined in Eqs. (<ref>), we tabulate the SU(8) fermion representations in Tabs. <ref>, <ref>, <ref>.
        For the right-handed down-type quarks of _R^Ω^c, they are named as follows
        
        
              _R^1̇^c ≡d_R^c  ,  _R^2̇^c ≡s_R^c  , _R^V̇İİ^c ≡_R^'''''^c  ,  _R^V̇İİİ^c ≡_R^'''^c   , _R^İẊ^c ≡_R^''''^c  ,       _R^3^c ≡b_R^c  , _R^ IV ^c ≡_R^c  ,  _R^ V ^c ≡_R^''^c   , _R^ VI ^c ≡_R^'^c   .
        
        
        For the left-handed SU(2)_W lepton doublets of (_L^Ω , - _L^Ω ), they are named as follows
        
        
                 ( _L^1̇ ,  - _L^1̇)  ≡ (e_L , - ν_e L )  ,  ( _L^2̇ ,   - _L^2̇)  ≡( μ_L  , - ν_μ L )   ,         ( _L^V̇İİ ,  - _L^V̇İİ)  ≡ ( _L^'''' , - _L^'''' )   ,   ( _L^V̇İİİ , - _L^V̇İİİ  )  ≡ ( _L^''' , - _L^''' )  ,   ( _L^İẊ ,  - _L^İẊ ) ≡ ( _L^''''' ,  - _L^''''' )   ,        ( _L^ 3 , - _L^3) ≡ ( τ_L  , - ν_τ L) ,   ( _L^ IV  , - _L^ IV ) ≡ ( _L^'' , - _L^'' )  ,        ( _L^ V  , -  _L^ V ) ≡ ( _L  , - _L )   ,  ( _L^ VI  , - _L^ VI  ) ≡  ( _L^' , - _L^' )  .
        
        
        According to the flavor identifications in Tabs. <ref>, <ref>, <ref>, we find the following up-type quark mass matrix
        
        
        _u      =   1/√(2)( ccc  
        0      0      c_5 ζ_1 /√(2)
        
         c_4  ζ̇_2 /√(2)    0      c_5 ζ_2 / √(2)
        
           c_5 ζ_1 /√(2)     c_5 ζ_2/√(2)      Y_
        ) v_ EW ,
        
        
        from the renormalizable Yukawa coupling term of Y_28_F28_F70_H+H.c. and the d=5 direct Yukawa coupling terms in Eqs. (<ref>) and (<ref>).
        The down-type quark and the charge lepton mass matrices are found to be
        
        
              ( _d )_3× 3  = ( _ℓ^T )_3× 3   ≈   1/4( ccc
         ( 2 c_3 + Y_ d_ℬ  )ζ̇_3^'     ( 2 c_3  +  Y_ d_ℬΔ_2̇ ) ζ̇_3^'    0   
        
          ( 2 c_3  +  Y_ d_ℬΔ_1̇^' )  ζ̇_3       (  2 c_3  +  Y_ d_ℬζ_23 ^-2  ) ζ̇_3        0   
        
          0     0       Y_ d_𝒜ζ_23^-1ζ_1   
        ) v_ EW ,
        
        
        from the d=5 indirect Yukawa coupling terms in Eqs. (<ref>).
        
        
        
        
        
        
        
        § THE CONJECTURE ON THE TOP QUARK MASS IN THE FLAVOR-UNIFIED SU(N) THEORIES
        
        Here, we provide a conjecture on the top quark mass, which states
        
        
        
        a rank-2 chiral IRAFFS is always necessary in the flavor-unified SU(N) theories so that only the top quark obtains mass with the natural (1) Yukawa coupling at the EW scale.
        
        
        
        
        
        
        According to our current convention, the EW symmetry breaking must occur in the (N-4)-th and also the last stage of SU(N ≥ 5) → ... →_ SM→ SU(3)_c ⊗ U(1)_ EM.
        In a rank-2 chiral IRAFFS of 
        
        
        
            ( N- 4) × [ N  , 1 ] _F⊕  [ N  , 2 ]_F
        
        
        one can always write down a Yukawa coupling of
        
        
        
            Y_  [ N  , 2 ]_F⊗ [ N  , 2 ]_F⊗ [ N  , N-4 ]_H + H.c.  .
        
        
        According to Ref. <cit.>, a rank-k anti-symmetric Higgs field in the SU(N) theory can only achieve the symmetry breaking pattern of SU(N) → SU(N-k).
        Obviously, the Higgs field of [ N  , N-4 ]_H in the rank-2 chiral IRAFFS only contains component that can be responsible for the (N-4)-th symmetry breaking stage (the EWSB stage), no matter how previous symmetry breaking patterns were achieved.
        This specific component is the SM Higgs doublet.
        Since the [ N  , 2 ]_F is only left with massless 10_F of the SM [One can justify this by looking at the SM fermions denoted by underlines in Tab. <ref>. ], with all other vectorial components integrated out before the (N-4)-th symmetry breaking stage, the corresponding Yukawa coupling in Eq. (<ref>) must give rise to the top quark mass.
        In other words, one unique SM Higgs doublet can only originate from a rank-2 chiral IRAFFS that is guaranteed to couple with the top quark naturally at the tree level.
        
        
        
        
        
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