High-quality grand unified theories with three generations

We extend the unitary groups beyond the ${\rm SU}(5)$ and ${\rm SU}(6)$ to look for possible grand unified theories that give rise to three-generational Standard Model fermions without the simple repetition. By demanding asymptotic free theories at short distances, we find gauge groups of ${\rm SU}(7)$, ${\rm SU}(8)$ and ${\rm SU}(9)$ together with their anomaly-free irreducible representations are such candidates. Two additional gauge groups of ${\rm SU}(10)$ and ${\rm SU}(11)$ can also achieve the generational structure without asymptotic freedom. We also find these models can solve the Peccei-Quinn (PQ) quality problem which is intrinsic in the axion models, with the leading PQ-breaking operators determined from the symmetry requirement.


I. INTRODUCTION
Grand Unified Theories (GUTs) [1,2] were proposed to unify all fundamental interactions described by the Standard Model (SM). Aside from the aesthetic aspect of achieving the gauge coupling unification in its supersymmetric (SUSY) extension [3], it is pragmatic to conjecture the zeroth law of GUT, namely, a successful GUT could address all intrinsic SM puzzles and as many physical issues beyond the SM as possible, with all necessary but the minimal set of fields determined by symmetry. One such longstanding puzzle that has not been well answered is the existence of three generational SM fermions, as well as their mass hierarchies in the framework of GUTs 1 . In a seminal paper [15], Georgi suggested to extend the minimal SU(5) into larger simple Lie group SU(N ) (with N ≥ 7), and built his three laws of GUTs. Instead of simple repetition of a set of anomaly-free irreducible representations (irreps) for three times, it is argued that the three-generational structure arises from different anti-symmetric irreps of the SU(N ).
In this paper, we investigate the possible non-minimal GUTs beyond the SU(5) and SU(6) that can give rise to three-generational SM fermions. The number of generations can be easily obtained according to the counting method in terms of the SU(5) irreps as given in Ref. [15]. It turns out that the SM fermion generations n g can already become three or beyond for the SU(7) group already [16]. Historically, the number of the SM fermion generations n g was also considered to be beyond three [17]. Meanwhile, the direct searches for the fourth-generational quarks at the Large Hadron Collider (LHC) have already excluded this possibility [18][19][20]. Therefore, only the non-minimal GUTs with their anomaly-free irreps that lead to n g = 3 cases will be considered in our study. In Georgi's third law, he decided that no individual irrep of the GUT group should appear more than once. Accordingly, he found that the minimal GUT group that give rise to n g = 3 is SU (11), with a total number of 1, 023 left-handed fermions [15]. Obviously, the third law prevent the three-generational structure through the simple repetition of the anomaly-free irreps. However, this may be a too strong constraint and was not usually adopted in the later studies. In our discussions, we modify Georgi's third law in a different version proposed by Christensen and Shrock [21] in the study of the dynamical origin of the SM fermion masses. A different point of view can be made such that the global symmetries can usually emerge once the original third law was abandoned, such as in the SU(9) GUT [22]. This can be advantageous at least in two aspects. The first advantage is the emergent global symmetry, with its breaking, can be a mechanism to explain the lightness of the Higgs boson, as was discussed by Dvali [23] in the context of the SUSY SU (6). The other advantage is that the global U(1) symmetry can be identified as the Peccei-Quinn (PQ) symmetry [24] for the strong CP problem. The emergent PQ symmetry, together with both the gauge and the global symmetries, can usually constrain the mass dimensions of the PQ-breaking operators and lead to a highquality axion [22,[25][26][27][28][29]. Two recent examples include the axion from the SO(10) [30] and the SUSY SU(6) GUT [31].
The rest of the paper is organized as follows. In Sec. II, we review Georgi's guidelines of building the non-minimal GUTs that can lead to three generations of SM fermions without simple repetition. Some other relevant results of the gauge anomaly cancellation, the Higgs representations, and the PQ quality are also setup there. Sec. III is the core of this work. We analyze all possible SU(N ) GUTs (up to SU (11)) and their anomaly-free fermion contents that can lead to three generational SM fermions according to Georgi's counting. The PQ charge assignments to Higgs fields and the corresponding PQ-breaking operators will be presented. We summarize our results and make discussions in Sec. IV.

A. Lie group representations and Georgi's guidelines
To facilitate the discussion, we express the fermion representations under the SU(N ) GUT group in terms of the set of rank-k anti-symmetric irrep of [N , k] as follows with n k being the multiplicity. Obviously, k = 0 corresponds to the singlet representation, and k = 1 corresponds to the fundamental representation, and etc. The singlet representations contribute neither to the gauge anomaly, nor to the renormalization group equations (RGEs). Through out the discussion, we always denote the conjugate representation such that [N , k] = [N , N − k]. It will be also useful to use a compact vector notation of n ≡ (n 0 , ... , n N −1 ) .
For a given rank-k anti-symmetric irrep of [N , k], its dimension and trace invariants are From Cartan's classification, it is well-known that the only possible Lie groups for non-minimal GUTs beyond the SU(5) or SO (10) are Since the exceptional group of E 6 has fixed rank, it is impossible to consider further extensions. For any irrep under these Lie groups, one can always decompose it under the subgroup of the SU(5). For example, the fundamental representation of the SU(N ) can be decomposed as The decompositions of the higher irreps can be obtained by LieART [32]. For an SU(N ) GUT, its fermion contents can be generally decomposed in terms of the SU(5) irreps as follows The anomaly cancellation condition leads to the following relation to the multiplicities In Ref. [15], Georgi argued that the counting of the SM fermion generations is equivalent to the counting of the multiplicity of the residual SU(5) irreps of [5 , 2] ⊕ [5 , 4], which is n g = n 2 − n 3 = n 4 − n 1 .
Note that the counting of the SM fermion generations in Eq. (9) does not rely on the realistic gauge symmetry breaking patterns. Based on Georgi's counting, it turned out that any GUT with orthogonal groups larger than the SO(10) essentially leads to n g = 0. This can be understood by decomposing the 16-dimensional SO(10) Weyl fermions under the SU(5) as The Weyl fermions from larger orthogonal groups are always decomposed under the SO(10) in pairs of 16 F ⊕ 16 F , and this can only lead to n g = 0.
Georgi's third law requires that not any representation of [N , k] should appear more than once, which means n k = 0 or n k = 1 in Eq. (1). This leads to a consequence that no global symmetry can emerge from the corresponding fermion setup. Instead, we adopt an alternative criterion by Christensen and Shrock [21], namely, the greatest common divisor of {n k } is not greater than unity. Therefore, one can expect the global symmetry of for all irreps of [N , k ] that appear more than once. This can be viewed as a generalization of the global symmetry in the rank-2 anti-symmetric theory of SU(N + 4) by Dimopoulos, Raby, and Susskind (DRS) [33] . The U(1) components of the global symmetry (11) can be identified as the global PQ symmetry, which are likely to lead to high-quality axion [31]. In this regard, the modified criterion of the fermion assignments is likely to solve the long-standing PQ quality problem [22,[25][26][27][28][29] in the framework of GUT.

B. Gauge anomaly cancellation
To have an anomaly-free non-minimal GUT, we have to solve the following Diophantine equation with the N -dimensional anomaly vector [34,35] being The property that the anomaly of a given irrep and its conjugate cancel each other is apparent in Eq. (14). Also, the self-conjugate representations must be anomaly-free such that A([N , N 2 ]) = 0 for N being even. Thus, the anomaly vector can be expressed as In practice, one has to decompose the SU(N ) fermion representations from [N , 1] to N , [ N 2 ] under the SU(5) in order to count the generations.

C. The Higgs representations
Once the fermion contents are determined for a particular non-minimal GUT, the Higgs fields can be determined by the following criteria is always assumed at its first stage, which requires an adjoint Higgs field [36]. The other possible symmetry breaking of SU(N ) → SU(N − 1) (with N ≥ 6) at the first stage is very likely to lower the proton lifetime predictions, and thus bring tension with the current experimental constraint to the proton lifetime from the Super-Kamionkande [37].
2. All possible gauge-invariant Yukawa couplings, which also respect the global symmetry in Eq. (11), can be formed.
3. Higgs fields to achieve any intermediate symmetry breaking stages are necessary, where their proper irreps contain the SM-singlet directions.
4. Only the Higgs fields with the minimal dimensions are taken into account.
Before proceeding to the more realistic models, we display the Higgs fields in the SU(6) GUT as an example. Its minimal anomaly-free fermion contents and decomposition under the SU (5) are The [5 , 1], one of the [5 , 4], as well as two singlets of [5 , 0] (6) GUT has a global symmetry of SU(2) 6 F ⊗ U(1) PQ and is a one-generational model according to Georgi's counting. The gauge-invariant and SU(2) 6 F -invariant Yukawa coupling can be expressed as follows with the minimal set of Higgs fields. Of course, a 35-dimensional adjoint Higgs field is necessary to achieve the first-stage GUT symmetry breaking of SU 38,39]. It turns out that the VEVs from Higgs fields of

D. The asymptotic freedom (AF)
The GUTs with their earliest versions are usually asymptotically free above the unification scale. However, there was no definite answer whether the AF should be retained.
An alternative criterion is to have an asymptotic safe theory, which reaches a fixed point at the short distance [40,41]. In general, the analysis of the asymptotic safe theories involves the RGEs of gauge couplings, as well as Yukawa and Higgs self couplings. This can only be performed for individual theory by specifying the symmetry breaking patterns. In the non-minimal GUTs, the AF is likely to be violated since the trace invariants of the rank-2 and rank-3 anti-symmetric representations scale as Previously, this was also considered in the SU(11) model [42], but with only fermions taken into account. In our discussions below, we study the short-distance behavior for non-minimal GUTs up to SU(11), with their minimal fermion setup. It turns out the minimal models in SU(10) and SU(11) violate the AF, and thus careful analysis of their unification couplings and scales are necessary for these two cases. The one-loop β coefficients are obtained by including both fermions and Higgs fields as follows with κ = 1 (1/2) for Dirac (Weyl) fermions, and η = 1 (1/2) for complex (real) scalars. For the adjoint Higgs fields, we always consider them to be real for the non-SUSY case. The AF can be determined by whether b 1 < 0 or not.

E. The PQ quality and axion
The global PQ symmetry has an intrinsic problem known as the PQ quality [22,[25][26][27][28][29]. In general, global symmetries are not fundamental but arise with the underlying gauge theories. They are believed to be broken by quantum gravity effects in the form of the following dimension-2m + n operator The size of PQ-breaking is constrained such that the minima of the QCD effective potential induced by axion should satisfy a/f a 10 −10 , which leads to a PQ quality constraint of It turns out that the mass dimension in Eq. (19) should be d 9 in order to have a reasonable axion decay constant f a ∼ O(10 12 ) GeV without much fine tuning of the coefficient k [26][27][28] in Eq. (19). Without knowing underlying symmetry origin of the Φ field, there is generally no reason to forbid any PQ-breaking operators with d 9.
Previous studies of the axion in the GUT [43][44][45][46] were made in both SU(5) and SO (10), where the global PQ symmetry was introduced by hand. Therefore, the issue of PQ quality was still present. Recent discussions of the PQ quality problem in the frame of GUT include the SO(10) [30] and SU(6) [31] cases. In the SO(10) GUT, the author made use of the generational symmetry in the limit of vanishing Yukawa couplings. A dimension-9 gauge-invariant operator to produce a high-quality axion was found, which is made up of Higgs fields for the intermediate symmetry breaking. In the minimal SU(6) GUT, it already possesses a global DRS symmetry as in Eq. (11). With the SUSY extensions, the authors [31] found a dimension-6 operator that lead to a highquality axion. Therefore, it becomes suggestive that the GUTs beyond the minimal versions are likely to solve the PQ-quality problem, with their local and emergent global symmetries. In the context of GUTs, the PQ-breaking operators can be formed by Higgs fields that develop vacuum expectation values (VEVs) at both the electroweak (EW) scale of v EW and the PQ symmetry-breaking scale of f a . This further alleviates the PQ quality constraint in Eq. (20)  f a . A natural question one can raise is whether the PQ quality constraint can be generally satisfied in the non-minimal GUTs with n g = 3. Impressively, we find this generally holds with proper assignment of the PQ charges to the Higgs fields.
The probes of the axion rely on the axion-photon effective coupling of For the GUTs, there is a universal prediction to the factor E/N SU(3)c = 8/3. The color anomaly factor of N SU(3)c relates the axion decay constant with the associate symmetrybreaking scale as v SB = |2N SU(3)c | f a , and also determines the domain wall number as N DW = 2N SU(3)c . In practice, one does not need to derive the factor by analyzing the symmetry breaking patterns. Instead, this can be obtained by using the 't Hooft anomaly matching condition [47] of Notice that in our current study, the physical axion does not arise at the GUT scale of ∼ 10 16 GeV. Instead, it arises from the phases of Higgs fields that are responsible for the intermediate symmetry breaking scale, with necessary orthogonality conditions imposed. One such example can be found in the minimal SUSY SU(6) GUT [31]. We focus on the PQ-breaking operators in the non-minimal GUTs, while the constructions of the physical axion in the specific GUT model will be left for future work.

III. THE RESULTS
In this section, we obtain our results of the SU(N ) GUTs that lead to n g = 3. Examples include SU (7), SU (8), and SU(9) groups, where the AF can be achieved. We also find that the higher groups of SU(10) and SU(11) with their minimal irreps cannot achieve the AF condition. For each case, we also look for the possible gauge-invariant and PQ-breaking operators. With proper PQ charge assignment at the GUT scale, we show that the PQ quality problem can be generally avoided in each model.
A. The SU (7) For the SU (7) group, the anomaly vector in Eq. (13) reads The decompositions of the SU (7) irreps under the SU(5) are the following There are two possibilities for n g = 3, namely, Since the number of fermions in two cases only differ by less than 10, we determine to consider both possibilities. Note in passing, a recent study [48] suggests that the SU(7) model can be suppress the proton decay with the proper embedding of the SM fermions. The Higgs sector of two SU(7) models is determined by the fermions and the global symmetries in Eq. (25) as follows Here and below, we use the square brackets to denote the real adjoint Higgs fields for the GUT scale symmetry breaking. By using the fermions and Higgs fields in Eqs. (25) and (26), we find that b A 1 = −5 and b B 1 = − 55 6 . Thus both the SU(7)-A and the SU (7) We assign the PQ charges for all SU(7) fermions and Higgs fields in Tab. I. The PQ charges cannot be uniquely determined from the PQ neutrality of the Yukawa couplings (27). Therefore, we assign the PQ charges by removing the possible dangerous PQ-breaking operators with low mass dimensions. In the SU(7)-A, one may assign PQ(21 F ) = q 1 and PQ(35 F ) = q 2 . Accordingly, it is easy to find two following PQbreaking operators These two operators would better be PQ-neutral due to their mass dimensions, and this leads to q 1 = q 2 = 0. Similarly in the SU(7)-B and larger groups below, we find the corresponding PQ charge assignments.
The color and electromagnetic anomaly factors, and domain wall numbers according to Eq. (22) are given by for two models. The leading gauge-invariant PQ-breaking operators become 2 (30b) For the leading PQ-breaking operator in the SU (7) The decompositions of the SU(8) irreps under the SU(5) are the following The possibility for n g = 3 with the minimal anomaly-free fermion content is given by By using the fermions and Higgs fields in Eqs. (33) and (34), we find that b 1 = − 2 3 < 0. Thus the minimal SU(8) model is asymptotically free. The gauge-invariant Yukawa couplings are We assign PQ charges for all SU (8) The leading PQ-breaking operators in the SU(8) model is C. The SU (9) For the SU(9) group, the anomaly vector in Eq. (13) reads The decompositions of the SU(9) irreps under the SU(5) are the following The possibility for n g = 3 with the minimal anomaly-free fermion content is given by Another possibility with more fermions of [9 , 2] By using the fermions and Higgs fields in Eqs. (40) and (41), we find that b 1 = − 15 2 < 0. Thus the minimal SU(9) model is asymptotically free. The gauge-invariant Yukawa couplings are and we assign the PQ charges in Tab. III. Naively, the PQ charge of the 84 H can be arbitrary according to the Yukawa couplings (42). It turns out a dimension-3 PQ-breaking can arise, which is dangerous from the dimensional counting. Therefore, we determine that PQ(84 H ) = 0. The color and electromagnetic anomaly factors, and domain wall numbers according to Eq. (22) are given by SU (9) : The leading dimension-9 PQ-breaking operator in the SU(9) model is According to the dimension counting in Ref. [27], this is likely to produce a high-quality axion.
D. The SU (10) For the SU (10)  The possibility for n g = 3 with the minimal anomaly-free fermion content is given by 10 , 9] = 120 F ⊕ 45 F ⊕ 8 × 10 F , dim F = 245 , By using the fermions and Higgs fields in Eqs. (47) and (48), we find that b 1 = 223 6 . Thus the minimal SU(10) model is not asymptotically free. The gauge-invariant Yukawa couplings are −L Y = 10 F ρ 120 F 45 Hρ + 10 F ρ 45 F 120 Hρ and we assign the PQ charges in Tab  The possibility for n g = 3 with the minimal anomaly-free fermion content is given by We assign the PQ charges in Tab The leading PQ-breaking operators in the SU(11) model is found to be IV. CONCLUSIONS  We have studied the set of non-minimal GUTs that can lead to the observed three generational SM fermions according to Georgi's counting. With the origin of the generational structure, these models themselves can be appealing to answer the most puzzling question of the SM fermion mass hierarchies. Our results suggest four such models that achieve the AF property at short distances, and two more that may be considered with further studies. The results are summarized in Tab. VI. The other important feature of the non-minimal GUTs in our study comes from their global symmetries, which was also previously noted in the SU(6) model [23,31]. Though the SU(6) model enjoys a global DRS symmetry, it turns out the SUSY extension was inevitable in order to produce a high-quality PQ symmetry. In six non-minimal GUTs of the current study, the sizes of the PQ-breaking effects due to the quantum gravity are generally under better control due to the gauge symmetries and the associated global DRS symmetries. It is thus reasonable to expect the long-standing PQ quality problem can be avoided in non-minimal GUTs with n g = 3, where the emergent global DRS symmetries are general.
Obviously, we expect the following studies to be performed for specific models, which include: (i) the viable symmetry breaking patterns, (ii) SM fermion mass hierarchies and their mixings, (iii) the physical axion mass predictions and the related experimental searches. A recent study of the SU(6) toy model [39] suggests that the bottom quark and tau lepton masses can be naturally suppressed to the top quark mass through the seesawlike mass matrices with their heavy fermion partners. Such heavy fermion partners for both the down-type quarks and charged leptons are general in the non-minimal GUTs with n g = 3. Furthermore, the non-minimal GUTs with n g = 3 can naturally lead to multiple symmetry breaking scales between the Λ GUT and the EW scale. Altogether, we expect that the observed fermion mass hierarchies among three generations can be realized with the appropriate symmetry breaking pattern in the non-minimal GUT with n g = 3. Above all, one has to analyze the gauge coupling unifications for the viable symmetry breaking patterns and predicts the proton lifetime. Since our models possess several intermediate scales, this usually requires the two-loop RGEs, together with the matching conditions and mass threshold effects [49,50].