The global $B-L$ symmetry in the flavor-unified ${\rm SU}(N)$ theories

We study the origin of the global $B-L$ symmetry in a class of flavor-unified theories with gauge groups of ${\rm SU}(N\geq 6)$. In particular, we focus on the ${\rm SU}(8)$ theory which can minimally embed three-generational SM fermions non-trivially. A reformulation of the third law for the flavor sector proposed by Georgi is useful to manifest the underlying global symmetries. The 't Hooft anomaly matching and the generalized neutrality conditions for Higgs fields play the key roles in defining the $B-L$ symmetry. Based on the global $B-L$ symmetry, we count the Higgs fields that can develop the VEVs and the massless sterile neutrinos in the ${\rm SU}(8)$ theory. We also prove that a global $B-L$ symmetry can always be defined in any ${\rm SU}(N\geq 6)$ theory when it is spontaneously broken to the SM gauge symmetry.


Introduction
Grand Unified Theories (GUTs), with their original formulations based on the gauge groups of SU (5) [1] and SO (10) [2], were proposed to unify all three fundamental symmetries described by the Standard Model (SM) into one fundamental symmetry.A widely accepted experimental justification of various GUTs is the proton decay processes.Through the analyses of the corresponding d = 6 fourfermion operators, it was pointed out by Weinberg, Wilczek, and Zee [3,4], that a conserved global U(1) B−L symmetry1 should originate from the GUTs such as the SU (5).
Historically, an extension of the unified gauge group beyond the SU (5) was motivated by Georgi to achieve the flavor unification [5].The central idea is to avoid the repetitive generation structure in the flavor sector of the GUT.Accordingly, Georgi conjectured that the observed three-generational SM fermions are non-trivially embedded in irreducible representations (irreps) of a simple Lie group beyond the SU (5).This insight is profound when confronting with the latest LHC measurements of the SM Higgs boson via different production and decay channels [6,7].The longstanding flavor puzzle in the SM quark and lepton sector can be mapped into a question of how does a SM Higgs boson distinguish fermions in different generations so that hierarchical Yukawa couplings are generated.At the UV scale, the flavor-unified theories have the intrinsic property that three-generational SM fermions must transform differently [5,[8][9][10][11][12][13][14][15].Therefore, one can expect a resolution to the fundamental flavor puzzles within this class of flavor-unified theories.
The purpose of this paper is to study the origin of the global U(1) B−L symmetry in the context of the flavor-unified theories with gauge groups of SU(N ≥ 6).The 't Hooft anomaly matching condition [16] will be applied.Since the global U(1) B−L symmetry in the SM is non-anomalous in the sense that [G SM ]2 • U(1) B−L = 0, it is natural to expect a global U(1) T symmetry 2 originated from the underlying UV theory such that [SU(N )] 2 • U(1) T = 0.In such flavor-unified theories, the gauge symmetry undergoes several intermediate symmetry-breaking stages before the electroweak symmetry breaking (EWSB).By performing the 't Hooft anomaly matching condition to the global U (1) T symmetries defined at each stage, the massless fermions such as the sterile neutrinos in the spectrum, can be counted precisely.A separate condition that is to require the Higgs fields that can develop the vacuum expectation values (VEVs) for each symmetry-breaking stage are neutral under the corresponding global U(1) T symmetries.This is a generalization from the fact that the SM Higgs field in the minimal SU (5) theory is U(1) B−L -neutral.
The rest of the paper is organized as follows.In Sec. 2, we revisit the definition of the B −L symmetry in the SU (5) theory from the 't Hooft anomaly matching condition.Generalized definition of the nonanomalous global U(1) T charge assignments can be given for an SU(N ) chiral gauge theory with rank-k anti-symmetric fermions.In Sec. 3, we describe the minimal flavor-unified SU (8) theory with the same fermion contents as were given in Ref. [13].Since this class of theories were not quite studied since Georgi's early proposal [5], we reformulate what was known as the third law of the flavor unification through the definition of the irreducible anomaly-free fermion set (IAFFS).The global Dimopoulos-Raby-Susskind (DRS) symmetries [17] will emerge naturally.We focus on the decomposition of the Higgs fields and fermions under the reasonable symmetry-breaking pattern according to Ref. [18].In Sec. 4, we define the global B − L symmetry in the SU (8) theory based on the 't Hooft anomaly matching condition.The consistent global U(1) T neutrality conditions to the Higgs fields at different symmetry-breaking stages will be applied so that the global U(1) B−L symmetry will naturally appear when the theory flows to the electroweak (EW) scale.In Sec. 5, we generalize the discussion to an arbitrary SU(N ) theory.It turns out that a global U(1) B−L symmetry can always be defined as long as the SU(N ) is spontaneously broken into the subgroup of G SM .We summarize the results and discuss the related issues for theories of this class in Sec. 6. App.A is given to define the decomposition rules and the charge quantizations for the SU (8) theory based on the symmetry-breaking pattern in the current discussion. 2 The global B − L symmetry in the SU(5) theory Let us start from the minimal SU(5) theory [1], with the chiral fermions of where the generational indices are suppressed.The first way to view the global U(1) B−L symmetry is to consider the renormalizable Yukawa couplings of It is straightforward to find the following global U(1) T 2 charge assignment of The U(1) B−L symmetry can be defined by the linear combination of the U(1) T 2 symmetry and the U(1) Y symmetry as with the original convention of t 2 = +1.
Alternatively, we can view the global U(1) T 2 and the U(1) B−L symmetries from the 't Hooft anomaly matching condition [16].The SU(5) fermions in Eq. ( 1) enjoy the global DRS symmetries [17] of ( Both U(1) □ and U(1) 2 are anomalous by the SU(5) instanton.However, one can always find a linear combination U(1) T 2 of Eq. ( 5) such that the mixed gauge-global anomaly of [SU(5)] After the SU(5) GUT-scale symmetry breaking, we can further define the global U(1) B−L symmetry as Since the mixed gauge-global anomaly of [SU(5)] ) F , (8a) Here, the color-triplet scalar of Φ 3 is expected to obtain masses at the GUT scale, and the Φ SM is identified as the SM Higgs doublet.According to the definition in Eq. ( 7), we have the global U(1) B−L charges of The 't Hooft anomaly matching condition reads
A separate condition is to require that the SM Higgs doublet of Φ SM is U(1) B−L -neutral [19], which leads to ã2 = 4t 2 ã1 = 4t 2 from Eq. ( 9).With both conditions, the U(1) B−L charges in Eq. ( 9) become A choice of t 2 = + 1 5 recovers the conventional definition of the U(1) B−L charges in the SM.
3 The flavor-unified SU(8) theory and its symmetry-breaking pattern

Overview
The flavor-unified theories based on the SU(N ) groups3 were first proposed by Georgi [5], where he required no repetition of any irrep in his third law, which reads Law (Georgi) no irreducible representation should appear more than once in the representation of the left-handed fermions.
The minimal solution is an SU (11) theory with 561 chiral fermions.The motivation of the third law in a flavor-unified theory is to avoid the simple repetitive structure of one generational SM fermions.Some of the examples that can lead to three-generational SM fermions by relaxing Georgi's third law can be found in Refs.[8-10, 13, 20-22].In this class of theories, only the anti-symmetric representations of the SU(N ) fermions, denoted as [N , k] F , will be considered, so that no SU(3) c /U(1) EM -exotic fermions can be present in the spectrum.The current LHC experiments [6,7] have confirmed the facts that the 125 GeV SM Higgs boson (i) only couples to the top quark with the natural ∼ O(1) Yukawa coupling, and (ii) can also distinguish lighter SM fermions from different generations so that the suppressed and hierarchical Yukawa couplings are generated.The flavor-unified theories have the property that both experimental facts of the SM Higgs boson can be naturally explained without extra flavor symmetries introduced by hand.For example, we have found that the SM Higgs boson from the 35 H in a twogenerational SU (7) toy model [15] only couples to the top quark at the tree level.
Since the class of the flavor-unified theories were not much studied, we wish to reformulate Georgi's third law [5].For our purpose, let us generalize the concept of "generation" into the concept of the IAFFS as follows.
Definition An IAFFS is a set of left-handed anti-symmetric fermions of R m R F L (R), with m R being the multiplicities of a particular fermion representation of R. Obviously, the anomaly-free condition reads R m R Anom(F L (R)) = 0. We also require the following conditions to be satisfied for an IAFFS.
• The greatest common divisor (GCD) of the {m R } should satisfy that GCD{m R } = 1.
• The fermions in an IAFFS can no longer be removed, which would otherwise bring non-vanishing gauge anomalies.
• There should not be any singlet, self-conjugate, or adjoint fermions in an IAFFS.
Obviously, one generational SM fermions as well as their unified irreps of 5 F ⊕ 10 F in the SU(5) theory in Eqs. ( 8) form an IAFFS.Based on the definition of the IAFFS, we conjecture the third law 4 of Conjecture (Chen) only distinctive IAFFSs without repetition are allowed in the GUT.
The flavor sectors with 3 × 5 F ⊕ 10 F in the SU(5) theory or 3 × 16 F in the SO(10) theory do not satisfy the third law.Furthermore, we have also reported in Ref. [15] that the early suggestion of the three-generational SU(7) models [8] cannot satisfy the third law since the repetitive IAFFSs can be found therein.Accordingly, the SU(N ) chiral fermions can be partitioned into several distinctive IAFFSs as follows with Georgi also gave the rules of counting the SM fermion generations in such SU(N ) theories [5], which read as follows The decompositions of other higher-rank irreps can be obtained by tensor products.
• Count the multiplicity of each SU(5) irrep as ν 5 F and so on, and the anomaly-free condition must lead to ν 5 F + ν 10 F = ν 5 F + ν 10 F .
• The SM fermion generation is determined by 4 This third law is a weaker version of the original version by Georgi [5], in the sense that the current setup leads to three distinctive 10 F 's and three identical 5 F 's at the EW scale.Given the decomposition of 10 F = qL ⊕ uR c ⊕ eR c , one can find that both left-and right-handed components of up-type quarks transform differently, the left-handed (righthanded) components of down-type quarks (charged leptons) also transform differently.Altogether, three-generational SM quarks/leptons must be distinguishable.Georgi's solution of an SU (11) theory [5] lead to three different 5 F 's and 10 F 's. 5 We use the notion of {k} ′ in Eqs.(12) and (16) to indicate distinctive IAFFSs according to the third law.
The multiplicity difference between the 10 F and the 10 F from a particular rank-k IAFFS was found to be [15] The total number of generations at the EW scale can be determined by [5] By expressing the fermions in terms of the IAFFSs in Eq. ( 12), one can immediately identify the emergent global DRS symmetries [17] of when m k ≥ 2 6 .The emergent global DRS symmetries in the general rank-k anti-symmetric chiral gauge theories were previously acknowledged in Refs.[23,24].With any rank-k IAFFS, the global U(1) λ k ⊗ U(1) k symmetries are broken by the SU(N ) instanton.One can define a non-anomalous global U(1) T k symmetry such that the mixed anomaly of [SU(N )] 2 • U(1) T k vanishes.According to Refs.[23,24], one finds the general U(1) T charge assignments of The other non-vanishing global anomalies read With the U(1) T charge assignments in Eq. ( 17), we can further determine the U(1) T charges for the Higgs field in the rank-k IAFFS through the renormalizable Yukawa couplings of The U(1) T charges of the Higgs fields are the center of the SU(N ) group.Notice that the Yukawa coupling of . cannot exist for the odd-k case, since the anti-symmetric property of the chiral fermion [N , k] F leads this term to vanish.
The three-generational SU(8) theory has the following anomaly-free chiral fermions with the undotted/dotted indices for the rank-2 IAFFS and the rank-3 IAFFS, respectively.The Arabic numbers and the Roman numbers are used for the SM fermions and the heavy partner fermions, respectively.According to Refs.[5,13,15,22], we find the SU(8) theory with the fermions in Eq. ( 20) is the minimal flavor-unified theory where three-generational SM fermions are embedded non-trivially.The total number of generations can be determined by Eqs. ( 14) and (15).The global DRS symmetries [17] from the fermion contents in Eq. (20) are at the GUT scale.
The most general gauge-invariant Yukawa couplings are7 In the G DRS -invariant limit, the Yukawa couplings are reduced to Altogether, we collect the SU(8) Higgs fields as follows By partitioning the chiral fermions to the IAFFSs in Eq. ( 12), the Higgs fields in the SU(8) theory can be determined according to the global DRS symmetries, while this point was not previously observed in Ref. [13].The adjoint Higgs field of 63 H is real and only responsible for the GUT scale symmetry breaking, while the others are complex.

The symmetry-breaking pattern
The symmetry-breaking pattern defines a realistic GUT from its UV setup to the low-energy effective theories at different scales.It was known that the Higgs representations are responsible for the gauge symmetry-breaking patterns, as well as the proper choices of the Higgs self couplings [18].For this purpose, we tabulate the symmetry-breaking patterns for the SU(N ) groups with various Higgs representations in Tab. 1.

Higgs irrep
Table 1: The Higgs representations and the corresponding symmetry-breaking patterns for the SU(N) group.
We consider the following symmetry-breaking pattern of the SU(8) GUT SU( 8) At the GUT scale, the maximal symmetry-breaking pattern of SU( 8) −→ G 441 can be achieved [18] due to the VEV of the SU(8) adjoint Higgs field as follows This symmetry-breaking pattern contains three intermediate scales between the GUT scale of v U and the EWSB scale of v EW .We denote two extended weak symmetries above the EW scale as SU(4) W ⊗ U(1) X 1 and SU(3) W ⊗ U(1) X 2 , since there can be both the left-handed and the right-handed quark/lepton multiplets under the corresponding extended weak symmetries through the decompositions.

Decompositions of the SU(8) Higgs fields
Below the GUT scale, all other intermediate symmetry-breaking stages will be due to the VEVs of the Higgs fields from the gauge-invariant Yukawa couplings of the SU(8) theory.During the symmetrybreaking stage of G → H, only the Higgs fields that contain the H-singlet components are likely to develop the VEV for this symmetry-breaking stage.We decompose Higgs fields in Eq. ( 22) into components that can be responsible for the sequential symmetry-breaking pattern in Eq. (25).The corresponding decomposition rules are defined in App. A. For Higgs fields of 8 H ,λ , they read For Higgs fields of 28 H , λ, they read For Higgs field of 56 H , they read For Higgs field of 70 H , they read Higgs The Higgs fields and their symmetry-breaking directions in the SU(8) GUT.The ✓and ✗represent possible and impossible symmetry breaking directions for a given Higgs field.
All components that are likely to develop VEVs for the sequential symmetry-breaking stages are framed with boxes, with their original components in the UV stages marked with underlines.Other SU(3) c -exact and/or U(1) em -exact components are neglected after the symmetry-breaking stage of G 441 → G 341 .Accordingly, we mark the allowed/disallowed symmetry-breaking directions for all Higgs fields in Tab. 2. At this point, these components are only determined by whether they contain the singlets of the unbroken subgroups.From the decomposition in Eq. ( 30), we found that the 70 H only contains the EWSB components of (1 , 2 , + 1  2 ) ′′′ H and (1 , 2 , − 1 2 ) ′′′ H , while all other Higgs fields contain components for the intermediate symmetry-breaking stages.Naively, one may expect both components of (1 , 2 , + 1  2 ) ′′′ H and (1 , 2 , − 1 2 ) ′′′ H to develop the VEVs of the same order.This is distinctive from the two-generational SU(7) model [15], where the Higgs field of 35 H from the rank-2 IAFFS therein only contains one EWSB component.As we shall show by analyzing their generalized U(1) T symmetries, only the (1 , 2 , + 1  2 ) ′′′ H can develop the EWSB VEV.Thus, we expect only the component of (1 , 2 , + 1  2 ) ′′′ H ⊂ 70 H that corresponds to the SM Higgs doublet at the EW scale.Also, the Higgs field of 56 H contains both the G 331 → G SM breaking component and the EWSB components.Through the definition of the generalized U(1) T symmetries, none of these components can actually develop any VEV since they are always charged.
For our later discussions of the symmetry-breaking pattern in the SU(8) theory, we denote the minimal set of Higgs VEVs as follows according to the Higgs decompositions in Eqs. ( 27), (28), and (30).

Decompositions of the SU(8) fermions
By following the symmetry-breaking pattern in Eq. ( 25), we tabulate the fermion representations at various stages of the SU(8) theory in Tabs.3, 4, and 5.All charges are obtained according to Eqs. (97a), (97b), (97c), and (97d).According to the counting rule by Georgi [5], it is straightforward to find n g = 3 in the current setup.Notice that the original counting rule was performed by decomposing all SU(8) irreps into the SU(5) irreps, and one may ask if the rule is valid when the symmetry-breaking pattern considered in Eq. ( 25) has no subgroup of SU (5).Explicitly, our decompositions in Tabs.3, 4, and 5 confirm that the counting rule is independent of the symmetry-breaking pattern.All chiral fermions are named by their irreps of the G SM .For the right-handed quarks of D Λ R c , they are named as follows SU( 8) T for the left-handed heavy lepton doublets.All left-handed neutrinos of ŇL are sterile neutrinos that do not couple to the EW gauge bosons.For the left-handed SU(2) W lepton doublets of (E Λ L , −N Λ L ) T , they are named as follows Through the analyses below, we shall see that all heavy (D Λ , E Λ , N Λ ) (with Λ = IV , . . ., İ X and named by Gothic fonts in Tabs. 4 and 5) acquire vectorlike masses during the intermediate symmetrybreaking stages.For the left-handed sterile neutrinos of ( Ň Λ L , Ň Λ ′ L , Ň Λ ′′ L ), four of them become massive with their right-handed mates during the intermediate symmetry-breaking stages in Sec. 4 and they are named as follows As we shall show in Sec. 4, the 28 F contains the third-generational fermions of 10 F , while the firstand second-generational fermionic components of 10 F 's are from the 56 F .The same result was also observed in Ref. [13].Since we are not analyzing the mass hierarchies of the SM quarks and leptons in the current paper, we only listed the possible names for the first-and second-generational fermionic components in Tab. 5.
4 The global B − L symmetry in the SU(8) theory By following the 't Hooft anomaly matching approach described in Sec. 2, we generalize the approach to the three-generational SU(8) theory.The generalization means two following requirements: 1. the generalized U(1) T symmetries should satisfy the 't Hooft anomaly matching condition, 2. the Higgs fields that can develop the VEVs for the specific symmetry-breaking stage should be neutral under the generalized U(1) T symmetries defined at that stage.
Table 6: The U(1) T 2 ⊗ U(1) T 3 charges for all massless fermions and Higgs fields in the SU(8) theory.
At the zeroth stage, no fermion in the spectrum obtain its mass.We define the U(1) T ′ 2 ⊗ U(1) T ′ 3 charges at this stage as The charge assignments are explicitly listed in Tab. 7. Accordingly, we find the following global anomalies of By matching with the global anomalies in Eq. ( 35), we find that At this stage, the c2 and d2 are not determined yet.
The Yukawa coupling between 8 F λ and 28 F can be expressed in terms of the G 341 irreps as follows They lead to the following fermion masses with the DRS limit of the Yukawa couplings.Without loss of generality, we choose the massive antifundamental fermion to be λ = IV at this stage.Thus, we can identify that (1 After this stage, the remaining massless fermions expressed in terms of the G 341 irreps are the following Fermions that become massive at this stage are crossed out by slashes.From the Yukawa couplings in Eq. ( 40), no components from the 56 F obtain their masses at this stage.Loosely speaking, we find that only one of the massive 8 F Λ is integrated out from the anomaly-free conditions of [SU(3 IV .One can further find that the remaining massless fermions from three 8 F λ and the 28 F form an IAFFS, and the remaining massless fermions from five 8 F λ and the 56 F form another IAFFS. Thus, we define the U(1) T ′′ 2 ⊗ U(1) T ′′ 3 charges at this stage as and they are explicitly listed in Tab. 8 for massless fermions and possible symmetry-breaking Higgs components.Accordingly, we find the following global anomalies of from the massless fermions in Eq. (42).By matching with the global anomalies in Eq. (37), we find that Two other coefficients of c′ 2 and d′ 2 are not determined yet.
The Yukawa coupling between 8 F λ and 28 F can be expressed in terms of the G 331 irreps as follows ) They lead to fermion masses of Without loss of generality, we choose λ = κ = V for the DRS limit in Eq. ( 23) at this stage.Thus, we can identify that (1 49b) is a mass mixing term, since the right-handed ň′′ c R has already obtained the Dirac mass with its left-handed mate in Eq. (41a).
The Yukawa coupling between 8 F λ and 56 F can be expressed in terms of the G 331 irreps as follows They lead to fermion masses of Without loss of generality, we choose λ = κ = VII for the DRS limit in Eq. ( 23) at this stage.Thus, we can identify that (1 A renormalizable Yukawa coupling between two 56 F 's cannot be present in Eq. ( 22).If no further Yukawa couplings were possible, the vectorlike fermions of (E , u , d , U) from the 56 F will be massless in the spectrum.It was suggested in Ref. [13] to consider the following d = 5 operator ) This d = 5 bi-linear fermion operator obviously breaks the global DRS symmetries in Eq. ( 21), and is only due to the gravitational effect8 .
charges for massless fermions and possible symmetry-breaking Higgs components in the G 331 theory.With the 't Hooft anomaly matching condition in Eq. ( 56) and the neutrality conditions in Eq. ( 57), the U(1) T ′′′ 2 ⊗ U(1) T ′′′ 3 charges are displayed in the parentheses.
The Yukawa couplings between the 8 F λ and the 28 F can be expressed as follows ) ) H ,λ + H.c.
They lead to fermion masses of where we choose λ = VI at this stage.Thus, we can identify that (1 , 2 , − 1 2 ) VI F ≡ (e ′ L , −n ′ L ), and 60) is a mass mixing term of sterile neutrinos.
The Yukawa couplings between the 8 F λ and the 56 F can be expressed as follows ) ) H , λ⟩ + H.c.

) ′′′ F ⊕ h h h h h h h h h h h
) F

F ⊕ h h h h h h h h h h h
The fermions that become massive at this stage are further crossed outs.After this stage of symmetry breaking, there are three generations of the SM fermion irreps, together with twenty-three left-handed massless sterile neutrinos.The third-generational SM fermions are from the rank-2 IAFFS of 8 F λ ⊕ 28 F , while the first-and second-generational SM fermions are from the rank-3 IAFFS of 8 F λ ⊕ 56 F .The flavor indices for the massive anti-fundamental fermions at this stage are chosen to be λ = VI and λ = ( İIX , İ X), respectively.
Thus, we define the U(1) T ′′′′ 2 ⊗ U(1) T ′′′′ 3 charges after this symmetry-breaking stage as and they are explicitly listed in Tab. 10 for massless fermions and possible symmetry-breaking Higgs components.Accordingly, we find the following global anomalies of  According to the generalized neutrality condition, the Higgs field of 56 H cannot develop any VEV through the symmetry-breaking pattern described in Eq. (25).Thus, the Higgs field of 56 H and the possible Yukawa coupling of 28 F 56 F 56 H + H.c. should be absent.

The EWSB stage
As we have previously mentioned, the EWSB is expected to be achieved by two components of (1 , 2 , + which precisely gives rise to the top quark mass with the natural Yukawa coupling of Y T ∼ O(1) at the tree level.The same result was also obtained in the two-generational SU(7) toy model [15].Based on this fact, we recover the observation that the 28 F only contains the 10 F for the third-generational SM fermions, while the 56 F only contains the 10 F 's for the first-and second-generational SM fermions [13].
Alternatively, let us consider the hypothetical situation where only the component of (1 , 2 , − 1 2 ) ′′′ H developed the EWSB VEV, one has The existence of massless sterile neutrinos in the flavor-unified theories with gauge groups of SU(N ≥ 6) is a generic feature.On the other hand, it is widely known that very small neutrino masses can originate from various seesaw mechanisms [29][30][31][32][33][34][35] as well as the d = 5 Weinberg operator [3].The 't Hooft anomaly matching condition to the generalized global U(1) T symmetries at different stages is useful to count the numbers of massless sterile neutrinos precisely, without knowing the possible mechanism of generating their masses.The SU(8) fermion sector contains twenty-seven left-handed sterile neutrinos from nine 8 F Λ , while only four of them obtain Dirac masses with their right-handed mates.The remaining twenty-three left-handed sterile neutrinos cannot obtain masses above the EW scale through the 't Hooft anomaly matching condition.With this fact, one can expect three-body decay modes of massive vectorlike fermions with two SM fermions plus one sterile neutrinos as their final states.Some of the relevant flavor-changing charged and neutral currents in the G 331 theory have been obtained in the two-generational SU(7) toy model, which can be automatically generalized to the three-generational case10 .

Discussions
Our current discussions focus on the minimal flavor-unified SU( 8 The corresponding global non-anomalous DRS symmetries at the GUT scale read The SU(8) fermion representation of 8 F Λ under the G 441 , G 341 , G 331 , G SM subgroups for the three-generational SU(8) theory, with Λ = (λ , λ).Here, we denote D Λ R c = d Λ R c for the SM right-handed down-type quarks, and D Λ R c = D Λ R c for the right-handed down-type heavy partner quarks.Similarly, we denote L Λ L = (e Λ L , −ν Λ L ) T for the left-handed SM lepton doublets, and 12 ) ′ F according to Tab. 5, one thus expect the identical T ′′ 3 charges of (3 , 6 , + 1 6 ) F and (3 , 4 , − 1 12 ) ′ F such that the first-and second-generational left-handed quark doublets carry the same B − L charges.This condition determines the coefficient of d2 = −4t 3 ,

3 )Table 10 : 2 ⊗
The U(1) T ′′′′ U(1) T ′′′′ 3 charges for massless fermions and possible symmetry-breaking Higgs components in the SM.The U(1) B−L charges are displayed in the parentheses.With a well-defined global U(1) B−L symmetry after the G 331 symmetry-breaking stage, one finds that all possible symmetry-breaking components from the 56 H are charged under the global U(1) B−L .
One may wonder why two anti-fundamental fermions of 8 F λ 1 2 ) ′′′ H ⊕ (1 , 2 , − 1 2 ) ′′′ H ⊂ 70 H purely from the decomposition of the Higgs irreps in Eq. (30).From their distinctive T ′′′′ 2 charges in Tab. 10, only one of them can actually develop the EWSB VEV with the U(1) B−L -neutrality condition.If one assumes that only the component of (1 , 2 , + 1 2 ) ′′′ This is consistent with the requirement of a well-defined global U(1) B−L symmetry for the thirdgenerational SM fermions.The corresponding Yukawa coupling reads Y T 28 F 28 F 70 H + H.c.