A two-generational ${\rm SU}(7)$ model with extended weak sector: mass hierarchies, mixings, and the flavor non-universality

We study a possible gauge symmetry breaking pattern in an ${\rm SU}(7)$ grand unified theory, which describes the mass origins of all electrically charged SM fermions of the second and the third generations. Two intermediate gauge symmetries of ${\cal G}_{341}\equiv {\rm SU}(3)_c \otimes {\rm SU}(4)_W \otimes {\rm U}(1)_{X_0}$ and ${\cal G}_{331}\equiv {\rm SU}(3)_c \otimes {\rm SU}(3)_W \otimes {\rm U}(1)_{X_1}$ arise above the electroweak scale. SM fermion mass hierarchies between two generations can be obtained through a generalized seesaw mechanism. The mechanism can be achieved with suppressed symmetry breaking VEVs from multiple Higgs fields that are necessary to avoid tadpole terms in the Higgs potential. Some general features of the ${\rm SU}(7)$ fermion spectrum will be described, which include the existence of vectorlike fermions, the tree-level flavor changing weak currents between the SM fermions and heavy partner fermions, and the flavor non-universality between different SM generations from the extended weak sector.


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). Though being successful in achieving the gauge coupling unification, one has to acknowledge that all existing unified models have not completely address some fundamental puzzles in the matter sector of the SM. In particular, the observed mass hierarchies and their mixings under the charged weak currents of three generational fermions have never been fully probed in the framework of GUT. A detailed description of the flavor puzzle can be found in recent Refs. [3,4]. All puzzles in the flavor sector originate from the three-generational structure. In most of the previous studies, one-generational anomaly-free fermion representations were trivially repeated with multiple times for the generational structure. For example, the minimal set of fermion contents are 3 × 5 F ⊕ 10 F in the SU(5) GUT, and 3 × 16 F in the SO(10) GUTs.
The flavor sector becomes more puzzling with the discovery [5,6] and the measurements [7] of the 125 GeV SM Higgs boson at the Large Hadron Collider (LHC). All existing measurements of the SM Higgs boson have confirmed its Yukawa couplings to the third generational fermions, both through the H → (bb , τ τ ) decays [8][9][10][11] and the ttH associated productions [12,13]. The ongoing searches are indicating the SM Higgs boson decays into the second-generational fermions of charm quarks [14,15] and muons [16,17]. These experimental results further strengthen the SM flavor puzzle into a SM flavor paradox. In the SM, each generation of fermions transform identically under the G SM ≡ SU(3) c ⊗ SU(2) W ⊗ U(1) Y . Therefore, a single SM Higgs boson is unlikely to generate hierarchical couplings to different flavors without additional symmetries. An intrinsic issue with the Higgs mechanism is what can be the natural Yukawa coupling. The proposal of the anarchical fermion mass scenario [18,19] will be adopted in the current discussion. From the renormalization group evolutions of the Yukawa coupling, it is unlikely to generate large hierarchies from O(1) to O(10 −6 ) when it evolves from the GUT scale to the electroweak (EW) scale. Studies of the running SM fermion masses can be found in Refs. [20][21][22][23][24][25]. From this point of view, the SM Higgs boson that is responsible for the electroweak symmetry breaking (EWSB) is most likely to give top quark mass, with the natural Yukawa coupling of y t ∼ O(1). Such a result was recently obtained in the third-generational SU(6) model [26]. Therefore, the question becomes what may be the underlying mechanism to generate all other suppressed SM fermion masses to the EW scale. Furthermore, three generational SM fermions exhibit mixings under the weak charged currents, which are described by the Cabibbo-Kobayashi-Maskawa (CKM) matrix [27,28] for the quark sector, and the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [29,30] for the lepton sector, respectively. The SM itself does not explain the origins of all measured fermion mixings. In the context of the minimal SU(5) GUT, Georgi and Jarlskog [31] realized natural mass relations between the downtype quarks and the charged leptons in three generations with extended Higgs fields of 3 × 5 H ⊕ 45 H .
Historically, a unification of the flavor sector was first made by Georgi [32], where he suggested that fermions in different generations may transform differently under the symmetries of the UV theory. It turns out this can be achieved in the framework of GUT, as long as one extends the gauge groups beyond the SU (5). Furthermore, one must include anti-symmetric irreducible representations (irreps) with rank larger than two to avoid the simple repetition of one generational anomaly-free fermions. This can be minimally achieved from a unified gauge group of SU (7), as was shown by Frampton [33,34]. Some early studies of the SU(7) models include Refs. [35][36][37][38][39][40]. Collectively, we dub the GUTs with gauge groups larger than SU(5) the non-minimal GUTs. Besides of some brief constructions of the fermion and Higgs spectrum, the non-minimal GUT was never studied in details from the physical perspective since its inception. A separate observation is that the non-minimal GUT groups are usually broken to the SM gauge group via several intermediate scales. For example, the third-generational SU(6) GUT undergoes two symmetry breaking stages of SU(6) → SU(3) c ⊗SU(3) W ⊗U(1) X → G SM [26,41]. During the intermediate symmetry breaking stages, some vectorlike fermions in the spectrum can obtain masses and also mix with the SM fermions. The existence of some Higgs mixing operators is likely to generate EWSB vacuum expectation values (VEVs) that contribute to light SM fermion masses, such as bottom quark and tau lepton in the third generation [26]. Based on the previous studies to the SU(6) toy model, we propose that Conjecture The realistic GUT can give arise to the observed mass hierarchies as well as the weak mixings of the SM fermions through its realistic symmetry breaking pattern, with the natural Yukawa couplings of O(1).
A separate feature of the non-minimal GUT is the automatic emergence of the global symmetries, as long as one relaxes the Georgi's third law [32] in formulating the anomaly-free fermion irreps. As was first observed by Dimopoulos, Raby and Susskind (DRS) [42], an anomaly-free SU(N + 4) gauge theory with N chiral fermions in the anti-fundamental irrep and one chiral fermion in the rank-2 anti-symmetric irrep enjoys a global DRS symmetry of G DRS = U(N ) = SU(N ) ⊗ U(1) , N ≥ 2 . (1) Notice that the global DRS symmetries in Eq. (1) come from the anomaly-free condition, and can be generally true with anomaly-free rank-k (k ≥ 3) fermion irreps. Though the original study of Ref. [42] deals with a strong interacting theory with the gauge symmetry broken through the bi-linear fermionic condensates, the emergence of the DRS global symmetries is valid when the underlying gauge symmetry is spontaneously broken by the Higgs mechanism. This result can be generalized to the non-minimal GUTs beyond the SU (5). Indeed, this was soon pointed out by Georgi, Hall, and Wise in Ref. [43], where they conjectured the Abelian component of the global DRS symmetry to be the automatic Peccei-Quinn (PQ) symmetry [44]. The emergent PQ symmetry from the non-minimal GUT may be an appealing source of another long-standing PQ quality problem [43,[45][46][47][48][49][50] in the axion models, e.g., a minimal construction may be realized in the supersymmetric SU(6) model with additional discrete Z 4 symmetry [41].
Motivated by our main conjecture, we study a two-generational SU(7) toy model. There are two major purposes of the current study, which are 1. to probe the origin of the SM fermion mass hierarchies and their mixings through the realistic symmetry breaking patterns; 2. to describe the gauge couplings of fermion sector in the non-minimal GUTs.
The rest of the paper is organized as follows. In Sec. 2, we describe the two-generational fermion content in the SU(7) model, and describe the gauge symmetry breaking pattern followed by SU(7) → SU(3) c ⊗ SU(4) W ⊗ U(1) X 0 at the GUT scale. In Sec. 3, we review some necessary ingredients from the third-generational SU (6) GUT. In particular, we focus on a gauge-invariant ν-term and its contribution to the Higgs potential. In Sec. 4, we discuss the Higgs sector of the SU (7) in details, with focus on the contributions from the gauge-invariant and DRS-invariant operators in the Higgs potential. These terms can be viewed as the generalized ν-terms in the SU(6) model. We found that some of such terms automatically generate a chain of additional VEVs through the tadpole-free condition in the Higgs potential. The gauge sector from the symmetry breaking pattern is described in Sec. 5. Sec. 6 is the core of this work, where we analyze the details of the realistic gauge symmetry breaking pattern. Based on the analyses, we obtain the fermion mass terms and mixings together with the Higgs VEVs described in Sec. 4. In particular, the analyses in this section will justify our identification of SM fermions in Tabs. 2, 3, and 4. In Sec. 7, we summarize the two-generational fermion masses, the quark mixings under the charged EW currents. Particularly, we display the flavor non-universality between different generational fermions through their gauge couplings to flavor-conserving neutral gauge boson Z µ . We summarize our results and make outlook in Sec. 8. In App. A, we give the decomposition rules and the charge quantization according to the desirable symmetry breaking pattern in the SU (7) GUT. All related Lie group calculations are carried out by LieART [51,52]. In App. B, we define the particle names and indices used for different gauge groups in the SU(7) model. In App. C, we give explicit derivations of the Higgs VEV generations through the complete set of Higgs mixing operators in the context of the SU (7).
We wish to make some disclaimers before presenting the main results. Throughout the current discussions, we focus on the mass origins and their mixings of two-generational SM fermions, while we do not address some general issues in the GUTs, such as (i) the gauge coupling unification, (ii) the proton lifetime, and (iii) the supersymmetric extension. We believe all these issues would better be studied in the realistic GUTs with n g = 3.
2 Motivation and the fermion sector in SU(7)

The flavor unification in GUTs
In a seminal paper [32] of the GUT model building including the flavor structure of the SM fermions, Georgi proposed three laws for any extension of the GUT group with the following chiral fermion content Here, [N , k] F represents a rank-k anti-symmetric irreps of the SU(N ) for the chiral fermions. It was also argued that only the [N , k] F should be considered in order to avoid the exotic fermions beyond the SM irreps in the spectrum. In the third law, he required no repetition of a particular irrep of [N , k] F . In other words, one can only have m k = 0 or m k = 1 in Eq. (2). Correspondingly, he found that the minimal possible GUT group is SU(11) with 561 chiral fermions in total.
The essence of Georgi's third law [32] is to prohibit the simple repetition of one set of anomaly-free chiral fermions in the flavor sector, such as 3 × 5 F ⊕ 10 F in the SU(5) model or 3 × 16 F in the SO(10) model. However, this was soon reconsidered by allowing the repetition of some fermion representations as long as the gauge anomaly is cancelled [33,34]. It is reasonable to conjecture that the third law for the realistic GUT building with multiple fermion generations can be modified by partitioning the SU(N ) chiral fermions into several irreducible anomaly-free sets as follows The anomaly factor for a generic rank-k anti-symmetric chiral fermion is given by [32,53,54] Anom Each set of anomaly-free fermions are composed of a rank-k (with 2 ≤ k ≤ N 2 ) anti-symmetric chiral fermion together with Anom([N , k] F ) copies of anti-fundamental chiral fermions. We propose that the third law of the flavor unification in GUT should be Conjecture Simple repetition of any irreducible anomaly-free fermion set in Eq. (3) is not allowed.
The notion of {k} in Eq. (3) represents the choices of the rank-k anti-symmetric fermion irreps without repeating itself. The conjectured third law thus leads to global DRS symmetries of This is certainly a generalization of the global DRS symmetry in the rank-2 anti-symmetric SU(N + 4) theory [42]. In the current SU(7) toy model, the specific choices of k in Eq. (3) will be required to reproduce n g = 2 according to the rules described below. In the realistic GUT, one should reproduce the observed n g = 3 for all SM fermions.
Georgi also gave the rules of counting the SM fermion generations, 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 It turns out that the multiplicity difference between 10 F and 10 F from a given irrep of [N , k] F can be expressed as Obviously for k = 2, one always has To avoid naive replication of generations, it is therefore necessary to consider anti-symmetric irreps with rank greater than or equal to 3. For the SU(6) case, the irrep of [6 , 3] F = 20 F is self-conjugate 1 and leads to ν 10 F [6 , 3] F − ν 10 F [6 , 3] F = 0 according to Eq. (6). In this regard, the SU (7) group is expected to be the leading GUT group that generates multiple generations non-trivially.
The SU(7) group was first considered by Frampton in Ref. [33], where he suggested the SU(7) group could generate multiple SM fermion generations. However, we wish to point out this was not true. To see this, we list two following sets of fermions from Ref. [33] {f L } SU (7) that can both lead to n g = 3 according to Georgi's rule. However, these fermions become if one partitions the fermions in terms of irreducible anomaly-free sets. Hence, there are trivial repetitions of one set of anomaly-free fermions for both cases suggested by Frampton. Through detailed analyses of the SU(7) symmetry breaking pattern below, we shall show that the SU (7) can never generate realistic SM fermion mass hierarchies for three generations.

Possible symmetry breaking patterns
Gauge symmetry breaking patterns are determined by Higgs representations, which was first studied in Ref. [55]. For this purpose, we tabulate the patterns of the symmetry breaking for the SU(N ) groups with various Higgs representations in table 1. One expects four different symmetry breaking stages for the SU(7) model. The zeroth-stage symmetry breaking occurs at the GUT scale, and one expects a maximally broken pattern of SU(7) → SU(3) ⊗ SU(4) ⊗ U(1) due to the Higgs VEVs of an adjoint Higgs field of 48 H . Since this stage leads to massive gauge bosons that can mediate the proton decays in the context of the minimal SU(5) GUT, it would be mostly proper to occur at the zeroth stage. However, one cannot determine whether the SU(4) or the SU(3) subgroup will describe the strong or weak sector. In the current discussion, we consider the following symmetry breaking pattern of the SU(7) This was previously considered in Refs. [35,38]. An alternative zeroth-stage symmetry breaking pattern of SU(7) → SU(4) s ⊗ SU(3) W ⊗ U(1) X 0 was previously discussed in Refs. [36,37,39,40,56]. We also wish to mention that such an ambiguity no longer arises when one considers the GUT groups of SU(8) [57][58][59] and SU(9) [34], where non-trivial embedding of three-generational SM fermions can be achieved. Given the SU(7) symmetry breaking pattern in Eq. (9), we define the decomposition rules and charge quantizations in App. A. Above the EWSB scale, the effective theory is described by a 331 model, which was previously studied in various aspects [26,41,].

The fermion content
The gauge anomaly factors of several leading SU (7) irreps are listed below according to Eq. (4). Any anti-symmetric irreps with higher ranks of the SU(7) are their conjugates. By decomposing the SU(7) fermions in terms of the SU(5) irreps, we have According to Ref. [32], one identifies one generational left-handed quarks from the 21 F , and 2 − 1 generational left-handed quarks from the 35 F . The two-generational SU(7) model contains the following fermions In other words, the fermions can be viewed by joining a rank-2 SU(7) model with a rank-3 SU(7) model. The corresponding global DRS symmetries are given by according to Eq. (5).
Given the global DRS symmetries in Eq. (13), we label the flavor indices as follows where the undotted and dotted indices are used to distinguish the SU (     We tabulate the SU(7) fermion spectrum in Tabs. 2, 3, and 4. The U(1) X 0 ,X 1 ,Y charges are obtained according to the assignments given in Eqs. (123a) and (123b). From the decomposition, we find one quark doublet of (3 , 2 , + 1 6 ) F from 21 F , two quark doublets of (3 , 2 , + 1 6 ) F and (3 , 2 , + 1 6 ) F , plus one mirror quark doublet of (3 , 2 , − 1 6 ) F from 35 F . The existence of the mirror quark doublet [91] is a distinctive feature from the previous one-generational SU(6) toy model [26], and they can emerge from non-minimal GUTs in general. According to the counting rule by Georgi [32], it is straightforward to find n g = 2 in the current setup. All chiral fermions are named by their SM irreps. For the right-handed quarks of (D Λ R ) c , they are named as follows For the left-handed lepton doublets of (E Λ L , −N Λ L ), they are named as follows All fermions are categorized according to their electrical charges in App. B. Names of all SM fermions will become transparent with the analyses of their mass origins from the symmetry breaking pattern in Sec. 6.
3 Review of the third-generational SU(6) toy model

The SU(6) model setup and the DRS symmetry
Before presenting the fermion masses in the two-generational SU(7) model, we review some necessary ingredients from the third-generational SU(6) toy model [26]. This was recently shown to split the bottom quark and tau lepton masses from the top quark mass in the third generation with natural Yukawa couplings of ∼ O(1). The minimal set of anomaly-free fermion content in the SU(6) model is given by This model enjoys a global DRS symmetry of In the Higgs sector of the SU(6) model, the Higgs components that can develop VEVs for the sequential stages of symmetry breaking after the GUT-scale symmetry breaking are These Yukawa couplings lead to massive vectorlike fermions from one copy of 6 F λ , and will be integrated out after the G 331 breaking. Accordingly, at least one of the (1 ,3 , − 1 3 ) H ,λ ⊂ 6 H ,λ should develop the VEV. We dub this the "fermion-Higgs matching pattern" for the symmetry breaking. When both of the (1 ,3 , − 1 3 ) H ,λ ⊂ 6 H ,λ develop VEVs for the G 331 breaking, there is still one copy of 6 F λ become massive at this stage. We dub this the "fermion-Higgs mismatching pattern" for the symmetry breaking [26]. Correspondingly, one naturally denote the VEVs of two (1 ,3 , − 1 3 ) H ,λ as
It turns out the SU(6) Higgs sector can naturally include a gauge-invariant mixing ν-term of According to Tab. 13, (ĩ ,j ,k) represent the fundamental/anti-fundamental indices of the SU(3) W group. This ν-term can also be DRS-invariant as long as p = −2q according to Tab. 5. The G SMsinglet terms that can develop the G 331 breaking VEVs correspond to theĩ = 3 components of two 3 ) H only contains the EWSB component withk = 1. Therefore, the above ν-term can lead to a following tadpole term in the Higgs potential We parametrize the G 331 -breaking and the EWSB VEVs as follows 2 The EWSB VEV of u φ will give masses to the bottom quark and the tau lepton. Hence, a hierarchy of u φ v 331 is expected. One can further demand no mass mixing term between the W ± and W ± . To see this, we write down the charged gauge boson masses from the SU(3) W ⊗U(1) X 1 covariant derivative in Eq. (62) by using the VEVs in Eq. (25), and they read It is straightforward to obtain the following orthogonal relation of λ=1 ,2 through the gauge transformations to Φ 3 ,λ , which assures the absence of mass mixing between the W ± and W ± . The VEV ratios in Eq. (26) are related as β =β − π/2. Thus, one can perform the following orthogonal transformations into the Higgs basis such that The ν-term in Eq. (24) will contribute to a VEV term of 1 to the Higgs potential. The minimization of the Higgs potential in Eqs. (22) leads to By equating Eqs. (31a) with (31d), and Eqs. (31b) with (31e), we have a relation of The solution of Eq. (32) naturally leads to u φ ∼ νv 331 GeV for the third-generational b quark and tau lepton masses in Eq. (32). This means some fine-tuning of the parameter ν in the SU(6) model is necessary [26].
The origin of the fine-tuning in the toy SU(6) model is simply because the ν-term is renormalizable. Such a fine-tuning problem is analogous to the µ-problem in the minimal supersymmetric Standard Model (MSSM) [92]. A widely accepted solution is the Kim-Nilles mechanism [93], where the µ-term in the MSSM superpotential is generated by non-renormalizable operator. Such operator can be possible with additional symmetries, for instance, the PQ symmetry. Analogously, we shall probe whether non-renormalizable operators that contribute to the Higgs VEV terms can emerge in the non-minimal GUTs with extended gauge symmetries beyond the third-generational toy SU (6). Obviously, the global DRS symmetries also differ between the two-generational SU(7) in Eq. (13) and the third-generational toy SU(6) in Eq. (18).

The Higgs sector and the VEV generations 4.1 The Higgs fields
The minimal set of Higgs fields can be obtained from the following SU(7) gauge-invariant Yukawa couplings 3 Names for Yukawa couplings will become manifest in the fermion mass generation. In the DRS limit, the Yukawa couplings become while other Yukawa couplings of (Y T , Y T C , Y C ) involve the DRS singlet fermions. Altogether, we collect the two-generational SU (7) Higgs fields as follows According to Ref. [55], the adjoint Higgs field of 48 H will be responsible for the GUT symmetry breaking of SU(7) → G 341 through its VEV of One can consider a generic Higgs potential for the adjoint Higgs field of 48 H as follows It turns out the desirable symmetry breaking of SU(7) → G 341 can be achieved with λ 2 > 0 [55]. The 48 H can be decomposed into two 3 × 3 and 4 × 4 diagonal blocks, plus two 3 × 4 off-diagonal blocks.
Obviously, the scalar components in two 3 × 4 off-diagonal blocks are the Nambu-Goldstone bosons for the massive gauge bosons (vectorial leptoquarks) at the GUT scale symmetry breaking. By combining the fermions in Eq. (12) and Higgs fields in Eq. (35), we tabulate their transformations and the most general charge assignments under the global DRS symmetries in Tab. 6. Before we analyze the details of the symmetry breaking, it will be useful to decompose all Higgs fields in Eq. (35) and to look for the singlet directions for the sequential symmetry breaking stages. The zeroth-stage symmetry breaking occurs at the GUT scale, which follows the pattern determined by the adjoint Higgs field and the charge quantization in Eq. (122). After the zeroth-stage GUT symmetry Table 7: The Higgs fields and their symmetry breaking directions in the SU(7) model. The and represent possible and impossible symmetry breaking directions for a given Higgs field. The represents the symmetry breaking direction that is mathematically possible, while can bring potential phenomenological constraints. The flavor indices for the Higgs fields that develop VEVs at different stages are also specified in the parentheses. breaking, they read with all possible Higgs VEV components responsible for the sequential symmetry breaking framed with boxes. All Higgs components without underlines or boxes are prohibited to develop VEVs so that the SU(3) c ⊗ U(1) EM remain exact. For our later convenience, we also give names to all Higgs fields that are responsible for the extended weak symmetry breakings. These components are determined by whether they contain the singlet directions according to the desirable symmetry breaking pattern in Eq. (9). to the so-called fermion-Higgs matching pattern. As will be described in Sec. 6, the number of Higgs VEVs will exactly match the copies of anti-fundamental fermions that acquire their vectorlike masses at each symmetry breaking stage. For each individual Higgs field, we assign VEVs for the highest symmetry breaking scale that is allowed by the symmetry breaking pattern given in Eq. (9). According to Tab. 7, we expect that the 7 H is mainly responsible for the G 341 symmetry breaking, the 21 H is mainly responsible for the G 331 symmetry breaking, and so on. Altogether, we arrive at the minimal set of Higgs VEVs for the sequential symmetry breaking stages as follows EWSB : It is natural to expect all these minimal set of VEVs at the particular symmetry breaking stage are of the same order, i.e., we have w 4 ,II ∼ w 4 and V 3 , Meanwhile, such an expectation are no longer valid for additional Higgs VEVs to be generated through the Higgs mixing operators below. Here, the flavor indices for all DRS-transforming Higgs fields will be chosen in accordance to the symmetry breaking pattern in Sec. 6.  (7) Higgs mixing operators and their U(1) 2 ,3 charges. The charge assignments with p 1 = q 1 = 0 and q 2 = − 1 2 p 2 = 0 are marked in parentheses.

The Higgs potential
The most generic Higgs potential for the with the renormalizable moduli terms expressed as  (7) Higgs mixing operators and their U(1) 2 ,3 charges. The and represent whether the specific operator can contribute to VEV terms in the Higgs potential or not. The charge assignments with p 1 = q 1 = 0 and q 2 = − 1 2 p 2 = 0 are marked in parentheses.
Obviously, the moduli terms are both gauge-invariant and DRS-invariant. For all mixing terms of O mix between Higgs fields, the DRS-invariance becomes non-trivial given that the possible operators may be DRS-charged. Here, we require that all renormalizable operators of O d≤4 mix listed in Tab. 8 to be DRS-neutral. For this reason, the only possible U(1) 2 ,3 charge assignments are In particular, three d = 4 operators of (O d=4 will play crucial roles in generating additional Higgs VEVs for the SM fermion masses. With the U(1) 1 ,2 charge assignments in Eq. (42), all non-renormalizable mixing operators of O d≥5 mix in Tab. 9 are inevitably DRS-charged. However, they can be allowed in the Higgs potential, in the sense that they can only be violated by the gravitational effects [45][46][47][48][49][50]94]. In Tabs. 8 and 9, we use the short-handed notations of Thus, we express the mixing terms in the Higgs potential as follows Obviously, only the Higgs VEV components should be taken into account in Eq. (40) when one minimizes the potential. This was previously pointed out in analyzing the explicit PQ-breaking terms by Ref. [95]. In App. C, we derive and determine whether a specific operator in Tabs. 8 and 9 can lead to VEV terms to the Higgs potential.  (7) Higgs fields.
For three renormalizable d = 4 operators that can lead to Higgs VEV terms, a suppression of induced EWSB VEVs are possible. Let us consider the generation of the suppressed VEV of u 2 ,3 . After the GUT scale symmetry breaking, the Higgs fields that lead to a VEV term in the operator of O d=4 . The Higgs potential must contain the following terms By minimizing the potential along the direction of the generated VEV u 2 ,3 according to Eq. (130), one finds a relation of µ 2 3 u 2 ,3 ∼ g 4A w 4 V 3 v t . By further assuming the natural relations of µ 3 ∼ w 4 and g 4A ∼ O(1), one finds that u 2 ,3 ∼ ( In addition to renormalizable d = 4 operators, we also find three non-renormalizable operators of (O d=5 A ) that can generate additional Higgs VEV terms.
Altogether, we end up with the following Higgs VEVs at each stage of symmetry breaking EWSB : The additional Higgs VEVs generated by the Higgs mixing operators in App. C are marked with boxes. We display the VEV generation chains through the series of operators in Fig. 1, with the minimal set of Higgs VEVs given in Eq. (39) as their input parameters. The EWSB VEV from the 21 H may lead to flavor-changing decay mode of t → h SM c, which should be further studied with the ongoing LHC searches for this rare decay mode. For our later usage, we denote the Higgs VEVs at different scales collectively as follows

The gauge sector
After the GUT scale symmetry breaking, the effective theory is described by an extended electroweak lead to seven and five massive gauge bosons, respectively. In this section, we describe the massive gauge bosons during two stages of symmetry breaking. Results obtained in this section will be used to describe the gauge couplings of fermions in Sec. 7. All group indices in this section follow the conventions defined in Tab. 13.

The SU(4) W ⊗ U(1) X 0 gauge bosons
We express the SU(4) W ⊗ U(1) X 0 covariant derivatives as follows for the SU(4) W fundamental representation. It becomes for the SU(4) W anti-fundamental representation. The SU(4) W generators of TĪ SU(4) are normalized such that Tr TĪ SU(4) TJ SU(4) = 1 2 δĪJ . The explicit form for the gauge fields of g 4W AĪ µ TĪ SU(4) + g X 0 X 0 I 4 X 0 µ can be expressed in terms of a 4 × 4 matrix as follows where we have defined a mixing angle of The notions of massive gauge bosons are determined according to their electric charges from the relation of [Q e (15 W ) , AĪ µ TĪ SU(4) ] = Q e AĪ µ TĪ SU (4) , withQ e (15 W ) defined in Eq. (126). Explicitly, we find the electrically charged gauge bosons of with the Higgs VEV of v 341 given in Eq. (47a). For the flavor-conserving neutral gauge bosons of (A 15 µ , X 0 µ ), the mass squared matrix reads 3 16 Obviously it contains a zero eigenvalue which corresponds to the massless gauge boson of X 1 µ after the SU(4) W symmetry breaking. The mass eigenstates can be diagonalized in terms of the mixing angle in Eq. (51) as follows The SU(4) W ⊗U(1) X 0 gauge couplings of (α 4W , α X 0 ) match with the SU(3) W ⊗U(1) X 1 gauge couplings as follows From the definitions of two mixing angles in Eqs. (51) and (63), we find a relation of The tree-level masses for seven gauge bosons at this stage read After the first-stage symmetry breaking, the remaining massless gauge bosons are (W ± µ , W ± µ ), (N µ ,N µ ), and (A 3 µ , A 8 µ , X 1 µ ).
In terms of mass eigenstates, the gauge bosons from the covariant derivative in Eq. (50) are expressed as As a consistent check, the (44)-component in Eq. (59) is reduced to − g X 1 4s θ G c θ G Z µ when setting X 0 = + 1 4 for the fundamental representation. Likewise, we find the explicit form the gauge fields of −g 4W AĪ µ (TĪ SU(4) ) T + g X 0 X 0 I 4 X 0 µ for the anti-fundamental representation as follows

The SU(3) W ⊗ U(1) X 1 gauge bosons
We express the SU(3) W ⊗ U(1) X 1 covariant derivatives for the SU(3) W fundamental and antifundamental representations as follows The λĨ SU (3) are Gell-Mann matrices, which are normalized such that Tr λĨ SU(3) λJ SU(3) = 2δĨJ . The gauge fields from Eq. (61a) can be expressed in terms of a 3 × 3 matrix with This stage of symmetry breaking is achieved by Higgs VEVs in Eq. (46b). The gauge boson masses from the Higgs kinematic terms are It is straightforward to obtain the mass eigenstates in terms of the mixing angle in Eq. (63) for this case The SU(3) W ⊗ U(1) X 1 gauge couplings of (α 3W , α X 1 ) match with the EW gauge couplings as follows From the definitions of two mixing angles in Eqs. (63) and (73), we find a relation of The tree-level masses for five gauge bosons at this stage read In terms of mass eigenstates, the gauge bosons from the covariant derivative in Eq. (62) become Likewise, the covariant derivative for the SU(3) W anti-fundamental representation gives

The SU(2) W ⊗ U(1) Y gauge bosons
We express the SU(2) W ⊗ U(1) Y covariant derivatives in the EW sector for the fundamental and anti-fundamental representations as follows with T I SU(2) ≡ 1 2 σ I . The gauge fields from Eqs. (71) can be expressed in terms of a 2 × 2 matrix with the Weinberg angle defined by 6 The symmetry breaking patterns and Yukawa couplings in the SU (7) In this section, we analyze the Yukawa couplings in the SU(7) symmetry breaking pattern. All fermions obtain their masses through the Yukawa couplings to the minimal set of Higgs fields given in Eq. (33) at each stage of symmetry breaking.

The G 341 → G 331 symmetry breaking
At the first stage, we consider the Higgs fields of (1 , 4 , − 1 4 ) H ,λ ⊂ 7 H ,λ and (1 , 4 , + 1 4 ) H ⊂ 7 H for the symmetry breaking, according to Tab. 7 and Eqs. (38). The Yukawa coupling term between the 7 F λ and 21 F is explicitly given by along the G 331 -singlet direction. The corresponding vectorlike fermion masses are with the VEVs in Eqs. (46)  The Yukawa coupling term between two 35 F 's is explicitly given by They give masses to the following vectorlike fermions In particular, the left-handed quark doublet of (3 , 2 , + 1 6 ) F and its mirror doublet of (3 , 2 , − 1 6 ) F from the 35 F become massive. Names of (s , c , U , E) become transparent according to their electrical charges, as well as their heavy masses. One of (1 , 1 , 0) λ F remains massless at this stage.
We find that one of the (1 , 3 , − 1 3 ) λ F ⊕ (3 , 1 , + 1 3 ) λ F ⊂ 7 F λ (chosen to be λ = II at this stage) become massive at this stage 4 . After integrating out the massive fermions, the residual massless fermions for the effective G 331 theory are

The G 331 → G SM symmetry breaking
The second-stage symmetry breaking is due to Higgs fields of (1 , 3 , The Yukawa coupling term between the 7 F λ and 21 F is explicitly given by along the G SM -singlet direction. They give masses to the following vectorlike fermions with the Higgs VEVs in Eqs. (46) and the DRS limit of Yukawa couplings in Eq. (34). Correspondingly, we can identify that (3 , 1 , The Yukawa coupling term between the 7 F λ and 35 F is explicitly given by along the G SM -singlet direction. They give masses to the following vectorlike fermions Correspondingly, we can identify that (3 , 1 , The Yukawa coupling term between two 35 F 's is explicitly given by along the G SM -singlet direction. They give the following fermion mass mixing terms Note that the 7 H has already developed a VEV for the G 341 symmetry breaking direction, and the VEV of V 3 is generated by Higgs mixing operator according to Fig. 1.
The Yukawa coupling term that mixes the 21 F and the 35 F is explicitly given by along the G SM -singlet direction. They give the following fermion mass mixing terms By analyzing the anomaly-free conditions for the effective G SM theory, we find the residual massless fermions of after this stage. The massive anti-fundamental fermions are chosen to be 7 Fİ ,I as in Eqs. (80) and (82).

The EWSB
Through the decomposition of all Higgs fields according to the symmetry breaking pattern in Tab. 7, we found that the 35 H only contains the EWSB direction. Similar situation also happened in the SU(6) toy model, where the Higgs field of 15 H that gives the top quark Yukawa coupling through the 15 F 15 F 15 H + H.c. contributes mostly to the EWSB [26]. Based on these facts, we conjecture that the EWSB is mostly achieved by Higgs field of (1 , 2 , + 1 2 ) H ⊂ 35 H , and the corresponding Yukawa coupling term in Eq. (33) leads to the top quark mass as follows Below, we list all other fermion mass terms from the additional EWSB VEVs generated from the Higgs mixing operators, as was displayed in Fig. 1.
The Yukawa coupling term between the 7 F λ and the 21 F can lead to along the EWSB direction. They give fermion masses of with λ = κ = 3 for the DRS limit of Yukawa couplings. We can identify that (3 , 1 , Obviously, the Eq. (90) gives common tree-level masses to the bottom quark and the tau lepton, which is the same as the prediction in the supersymmetric SU(5) Georgi-Glashow model and the third-generational SU(6) model [26].
The Yukawa coupling terms between the 7 Fλ and the 35 F can lead to along the EWSB direction. They give fermion masses of Eq. (92a) gives common tree-level masses to the sea quark and the muon. The other EWSB VEV from Eq. (92b) gives mass mixing terms between the second-generational fermions and heavy partner fermions.
The Yukawa coupling terms between two 35 F 's can lead to along the EWSB direction. They give fermion mass mixing terms of The Yukawa coupling term that mixes the 21 F and the 35 F in Eq. (85) can further become along the EWSB direction. They give fermion mass mixing terms of The SM fermion masses, mixings, and flavor non-universality In this section, we summarize the fermion mass spectrum according to the symmetry breaking patterns presented in Sec. 6. With the fermion identifications through the previous analyses, we also obtain their gauge couplings according to the gauge sector described in Sec. 5.

The quark mass spectra and their mixings
We start from the up-type quarks with Q e = + 2 3 in Eq. (127). By using the basis of (c , c , U , t), we find their mass matrix of from Eqs. (77), (84), (86), (88), (94), and (96). Several features of Eq. (97) should be observed. First, the charm and top quarks form the 2 × 2 mass matrix of which resembles the conjectured mass matrix by Georgi and Jarlskog [31]. Second, one finds that In the limit of the vanishing Yukawa mixing of Y T C → 0, the lightest charm quark must be massless. The masses of two heaviest vectorlike quarks and the top quark are approximately Thus, the charm quark mass can be approximately expressed as Obviously, the charm quark becomes massless when all generated EWSB VEVs of (u 2 , u 2 , u 2 ) and/or the mixing Yukawa coupling of Y T C vanish.
In general, any fermion mass matrix can be diagonalized in terms of bi-unitary transformation of We focus on the M F M † F in order to obtain the CKM mixing in the quark sector. The diagonalization of the up-type quark mass matrix in Eq. (97) can be performed in terms of perturbation expansion as follows The only contains the mass terms with the SU(4) W ⊗ U(1) X 0 -breaking and SU(3) W ⊗ U(1) X 1 -breaking VEVs, and can be diagonalized by the orthogonal transformation as follows where we approximated the matrix to the order of w −1 4 . By further including the terms from the , which depend linearly on the EWSB VEVs of (u 2 , u 2 , u 2 ), the mass eigenstates of the charm and the top quarks (denoted asĉ andt) are related to the gauge eigenstates as The mass matrices for the down-type quarks with Q e = − 1 3 and charged leptons are correlated. Here, we express their mass matrices in terms of the basis of ({s , s , S} , {b , D , D }) and ({µ , E , e } , {τ , e , e }) as follows Below, we focus on the down-type quark sector in order to address the electroweak mixing. We find that One can expand the down-type quark mass matrix in terms of the VEV hierarchies as follows The leading mass terms from the can be diagonalized by the orthogonal transformation as follows mass terms, we find the strange and bottom quark masses of at the tree level. The corresponding EWSB VEVs for their masses are generated from operators in Eqs. (130) Together with the left-handed quark mixing matrix in Eq. (111), we find the following approximation to the 2 × 2 CKM matrix Indeed, the current 2 × 2 CKM matrix resembles the observed feature of the realistic 3 × 3 CKM matrix, namely, it is almost diagonal. Notice that we have neglected all higher order correction terms suppressed by w −1 4 in Eq. (112). Given the mass matrices for the lepton sector in Eq. (106b), it is straightforward to infer the tree-level masses of This means the b − τ mass unification is extended to the s − µ mass unification at the tree level. In the context of the Georgi-Glashow SU(5) model, the possible b − τ mass splitting was attributed to the renormalization group effects [96]. However, results therein cannot be naively applied to the (b , τ ) and (s , µ) mass ratios in the non-minimal GUTs. To fully evaluate their mass splitting, we expect two prerequisites of: (i) the evaluation of the intermediate symmetry breaking scales from an appropriate GUT group, and (ii) the identification of the SM fermion representations with the extended color and weak symmetries above the EW scale. Both are distinctive features of the non-minimal GUTs, and we defer to analyze the details in the future work.

The neutrino masses
We also summarize the neutrino masses in the SU(7) toy model. According to our conventions in Eq. (127), the neutral fermions include two EW active neutrinos of (ν µ , ν τ ), three vectorlike massive neutrinos of (n , n , n ) from the SU(2) W -doublets, nine left-handed sterile neutrinos, and one vectorlike massive sterile neutrinos ofň. From Eqs. (75), (80), and (82), we find the vectorlike neutrino masses of m n = mň ∼ O(v 341 ) and m n ∼ m n ∼ O(v 331 ). All other EW active neutrinos and sterile neutrinos are massless from the tree-level Yukawa couplings. Neither can we find any tree-level Yukawa coupling that mixes the active neutrinos and the massive neutrinos in the current setup. Meanwhile, it is known that the loop-level effects can be ubiquitous in the neutrino mass generation [97] and it is most appropriate to take the analysis in the three-generational non-minimal GUTs.

The fermion gauge couplings with the extended weak symmetries and the flavor non-universality
We proceed to derive the fermion gauge couplings. Since the color symmetry of SU(3) c is always exact in the current context, we focus on the extended weak symmetries. Some general features of the fermion gauge couplings in the SU(7) model can be outlined. First, different SM fermion generations transform differently, as the current SU(7) model suggests through its fermion decompositions in Tabs. 2, 3, and 4. Consequently, we show manifestly that the flavor non-universality can be expected through the flavor-conserving neutral currents from the SU(4) W ⊗ U(1) X 0 symmetry breaking. Second, nonminimal GUTs such as SU(7) model and beyond contain vectorlike fermions in the spectrum. This can be manifestly confirmed through the flavor-conserving neutral currents from the SU(3) W ⊗ U(1) X 1 symmetry breaking in the current context. Third, the tree-level currents include both flavor-changing charged currents (FCCC) mediated by (W ± , W ± , W ± ), as well as flavor-changing neutral currents (FCNC) mediated by (N µ ,N µ , N µ ,N µ , N µ ,N µ ). The tree-level FCNCs in the current context never involve two different flavors of SM fermions. Instead, they only involve one SM flavor with another heavy vectorlike fermion with the same electric charge, as will be explicitly given in Eqs. (114b) and (116b). Below, we organize the relevant couplings according to the symmetry breaking stages described in the current context.

The SU(4) W ⊗ U(1) X 0 gauge couplings
After the first-stage symmetry breaking, the FCCC and the FCNC are expressed as Table 10: The couplings of the flavor-conserving neutral currents mediated by Z µ in the V − A basis, with θ G being the SU(4) W ⊗ U(1) X 0 mixing angle defined in Eq. (51).
The flavor-conserving neutral currents are expressed as follows in the V − A basis and we tabulate the vectorial and axial couplings of (g V f , g A f ) in Tab. 10. Manifestly, the gauge couplings of two-generational SM fermions with the flavor-conserving neutral boson of Z µ are distinctive, which is the source of flavor non-universality. This is consistent from what we found from the fermion irreps in Tabs. 2, 3, and 4. The flavor non-universality is only possible with the extended weak symmetries of SU(4) W ⊗ U(1) X 0 , or in the SU(7) and beyond non-minimal GUTs. As we have discussed previously in Sec. 2, the non-trivial embedding of multiple SM generations requires at least rank-3 (or above) anti-symmetric irreps that are not self-conjugate. One should also expect the flavor non-universality in a realistic non-minimal GUT from its extended weak and strong sectors.

The SU(3) W ⊗ U(1) X 1 gauge couplings
After the second-stage symmetry breaking, the FCCC and the FCNC are expressed as  The flavor-conserving neutral currents are expressed as follows in the V − A basis and we tabulate the vectorial and axial couplings in Tab. 11. The fermions of (c , U , s , D , e , E) only exhibit vectorial gauge couplings with the Z µ . This can be expected, since they already obtain vectorlike masses through the first-stage symmetry breaking, as one can find in Eqs. (75) and (77). The SM fermions with the same electrical charges couple to the Z µ universally. This can also be expected, since the SU(6) GUT, which can unify the G 331 gauge symmetries minimally, cannot have multiple fermions embedded non-trivially according to Georgi's counting rule and the third law. Therefore, the flavor universality of the SM fermions should be expected through the flavor-conserving neutral currents of the effective G 331 theory based on the non-minimal GUTs. Consequently, the gauge couplings for the first generational SM fermions should be identical to the second and the third generational SM fermions as we have listed in Tab. 11. This is distinctive from the fermion contents in several previous 331 model studies [63-70, 72-77, 80-84, 86, 88-90], where the flavor non-universalities were assumed at the beginning.

The electroweak gauge couplings
At the stage of the EWSB, the flavor-conserving neutral currents are expressed as follows in the and we tabulate the vectorial and axial couplings in Tab. 12. Consistent SM fermion gauge couplings are obtained. Besides of the SM fermion couplings, all heavy partner fermions in the spectrum only have vectorial gauge couplings to the Z µ boson.

The search for the Z µ gauge boson
The non-minimal GUTs such as the two-generational SU(7) model predict an extended G 331 effective theory above the EWSB scale according to the symmetry breaking pattern in Eq. (9). All massive gauge bosons and vectorlike fermions are expected to have masses of ∼ v 331 according to the previous analyses. According to Eqs. (116a), (116b) and (117), only the flavor-conserving neutral Z µ can couple to SM fermions and anti-fermions, while all other massive gauge bosons always mediate between a SM fermion and a heavy vectorlike fermion, or between heavy vectorlike fermions. Accordingly, we can mostly expect the current and/or future terrestrial collider searches for the Z µ . Based on the related gauge couplings in Tab. 11, we estimate the leptonic decay branching ratios of Br[Z µ → + − ] 331 = 0.2 by assuming that the Z µ cannot decay into the vectorlike fermions of (S , D , e , e ). The current LHC searched for the Z µ via the di-lepton final states and assumed the sequential SM (SSM) scenario [98] was assumed. The corresponding leptonic decay branching ratio reads Br[Z µ → + − ] SSM = 0.06 according to the gauge couplings in Tab. 12. A rescaling of the couplings leads to the signal strength of The ratio of the production cross sections is found to be where we estimate the ratio of [99] by using the MSTW2008PDF [100]. Altogether, we find that µ 0.57, which sets a limit of m Z 4.8 TeV to our current model setup according to the LHC searches for the SSM Z µ [98]. By further using the gauge boson mass in Eq. (68b) and the mixing angle relation in Eq. (67), we estimate that the current LHC searches have set a limit of v 331 12 TeV to the non-minimal GUTs.

Discussions
In this work, we initiate a study of the non-minimal GUTs with multiple generational SM fermions that transform differently in the UV theory. A two-generational SU(7) unified theory is a reasonable step towards more realistic model building. Remarkably, the anomaly-free fermion contents in the UV theory do not display any generational structure, as one can find in Eq. (12). It will be straightforward to find that the first non-minimal GUT with the minimal fermion contents that can lead to three generations at the electroweak scale has a unified group of SU(8) [57][58][59], which is also composed by both the rank-2 and the rank-3 anti-symmetric fermions, plus the anti-fundamental fermions. Regarding this, some of the results in the two-generational SU(7) model will be expected to be relevant in a more realistic model construction. Below, we summarize some major results of the current work.
Firstly, the non-minimal GUTs are built based on the conjectured third law of the flavor unification. Naturally, the unified gauge symmetry undergoes multiple intermediate stages of symmetry breaking below the GUT scale. In the current context, we focus on the symmetry breaking pattern where the weak symmetries are extended. It can also be expected that the realistic symmetry breaking pattern of a three-generational theory includes both extended strong and weak sectors beyond the SM gauge symmetries. At each stage of symmetry breaking, the vectorlike fermions that acquire masses are predictable through the anomaly-free conditions. Similar to the SU(6) model, we found that the Higgs field of 35 H can only develop VEV for the EWSB and give the top quark mass in the given symmetry breaking pattern. These results suggest that the fermion masses are acquired through the natural Yukawa couplings of Y f ∼ O(1) in the non-minimal GUTs.
Secondly, the emergent global DRS symmetries are generally expected with our conjectured third law on the anomaly-free fermion content. We determine the Higgs fields according to the minimal set of Yukawa couplings with the corresponding DRS symmetries. With the minimal set of Higgs VEVs in Eqs. (39), there can be additional Higgs VEVs generated through the Higgs mixing terms. They are responsible to give masses to other electrically charged SM fermions other than the top quark. In other words, the previously observed VEV generations through a d = 3 operator in the SU(6) model [26] are generalized in the current context. In the context of the SU(7) model, the additional Higgs VEVs are either generated from d = 4 renormalizable operators, or from d ≥ 5 non-renormalizable operators. To have a complete VEV generation chain as we have shown in Fig. 1 in the current model, the non-renormalizable operators are inevitably DRS-charged, and they can only be allowed with the gravitational effect that violates the global symmetries in general.
Thirdly, the observed fermion mass spectrum can partially explain the observed mass hierarchies of the SM fermion. In particularly, the charm quark mass can be reasonably suppressed from the top quark. Notice that Yukawa texture that leads to vanishing tree-level charm quark mass in Eq. (94) is purely due to the symmetries in the current setup. Besides, we have successfully displayed the observed CKM mixing pattern with two generations. These results may be viewed as positive hints for the future construction of the three-generational model. However, the current model predicts the unrealistic mass relations of m s ≈ m b and m µ ≈ m τ . As we have mentioned in the second point, such VEV generation chains may be completely due to the non-renormalizable operators with the gravitational effects. Besides, the hierarchical SM fermion masses among three generations were long conjectured due to the radiative mechanism with extended gauge bosons and/or Higgs fields [101][102][103], including in the class of the SU(N ) GUTs [57]. Altogether with the distinct gauge symmetries and the emergent DRS symmetries in the three-generational non-minimal GUTs, it is premature to conclude such degenerate fermion mass pattern among different generations. Of course, the usual b − τ mass unification is further extended to the second generational fermions of (s , µ) at the tree level. Thus, it will be reasonable to investigate the renormalization group effects [96] in a realistic non-minimal GUT, with two prerequisites of: (i) the identification of the intermediate symmetry-breaking scales, and (ii) the identification of the SM fermion representations with the extended color and weak symmetries.
Lastly, we wish to point out the early constructions of three-generational SU(7) model by Frampton [33,34] cannot be realistic models. Our previous argument was that the partition of the fermions into irreducible anomaly-free sets in Eqs. (8) violates our conjectured third law. According to the current analyses, the SU (7) can only undergo two more intermediate symmetry breaking stages above the EWSB scale. One can naturally expect that the first generational fermion masses, in particular the down quark and the electron, will be degenerate with SM fermions that carry the same electric charges.
In addition to the study of the SM fermion masses and their weak mixings, several new physics ingredients have automatically emerged in the current context, which include • Emergence of the DRS symmetries that may give rise to high-quality axion [43,[46][47][48][49]95] for the strong CP problem [44].
• The vectorlike mirror quark doublets [91] and the sterile neutrinos from the minimal fermion contents.
• Flavor-changing neutral currents and flavor non-universal gauge couplings [104] from the extended weak symmetries. In particular, the flavor non-universality originates from the non-trivial embedding of multiple generations into the unified theory.
These ingredients are expected to be general in non-minimal GUTs with three generational SM fermions, given their anomaly-free fermion content according to our conjectured third law of flavor unification, as well as the realistic symmetry breaking patterns from the group theoretical considerations.  B Name scheme of the SU (7) fermions Here, we list the names for fermions in Tabs. 2, 3, and 4 according to the symmetry breaking patten analyzed in Sec. 6. Names of all heavy partner fermions are expressed in Gothic fonts. According to Tabs. 2, 3, and 4, we name fermions according to their electric charges as follows Neutrinos marked withˇare sterile neutrinos, since they are SM singlets.
C The Higgs mixing operators in the SU(7) Higgs potential In Sec. 4, we argue that the VEV terms in the SU (7) Higgs potential should be determined through gauge-invariant operators listed in Tabs. 8 and 9. In this section, we show explicitly whether each term contributes to a VEV term or not according to the Higgs decompositions in Eqs. (38). This can be made by checking whether a specific operator composed by the Higgs components framed with boxes is gauge-invariant.

C.1 General rules
We list some general rules to obtain the VEV term contributions to the Higgs potential. They read as follows 1. One should always choose the Higgs VEV components, and look for gauge-invariant operators at each stage of symmetry breaking. In the current context, the VEV components are framed with boxes in Eqs. ( Obviously, none of the VEV components in Eqs. (128) are gauge-invariant after the decomposition.

C.3 The d = 4 operators
For the operator of O d=4 A , we find According to the minimal VEV assignments for the fermion-Higgs matching pattern in Eqs. (39), this operator leads to a VEV of u 2 ,3 to avoid the tadpole term.
Obviously, the gauge-invariant VEV term in Eq. (129) plays the similar role as the ν-term in the SU(6) model [26] and can lead to a tadpole term. By denoting the coefficient of this operator as g 4A , we have ν 4A = g 4A w 4 ,II . According to the study of the third-generational SU(6) model, one expects ν ∼ O(100) GeV. Similar to the SU(6) model, a fine-tuning is also observed. Thus, the ν-problem persists in the current context and will be solved in the realistic models.