More on Fake GUT

It is remarkable that the matter fields in the Standard Model (SM) are apparently unified into the $\mathrm{SU}(5)$ representations. A straightforward explanation of this fact is to embed all the SM gauge groups into a simple group containing $\mathrm{SU}(5)$, i.e., the grand unified theory (GUT). Recently, however, a new framework"fake GUT"has been proposed. In this new framework, the apparent matter unification can be explained by a chiral gauge group $G$, $ G \supset \mathrm{SU}(5)$. We emphasize that the SM matter fields are not necessarily embedded into the chiral representations to explain the apparent unification. In this paper, we discuss details of concrete realizations of the fake GUT model. We first study the model based on $\mathrm{SU}(5) \times \mathrm{U}(2)_H$, where $\mathrm{SU}(3)_c$ in the SM is from $\mathrm{SU}(5)$ while $\mathrm{SU}(2)_L\times \mathrm{U}(1)_Y$ are from the diagonal subgroups of $\mathrm{SU}(5) \times \mathrm{U}(2)_H$. We also extend this model to the one based on a semi-simple group, $\mathrm{SU}(5) \times \mathrm{SU}(3)_H$, so that $\mathrm{U}(2)_H$ is embedded in $\mathrm{SU}(3)_H$. We also show that this framework predicts rather different decay patterns of the proton, compared to the conventional GUT.


Introduction
In the Standard Model (SM), the matter fields are apparently unified into the SU(5) representations. In general, chiral fermions consistent with SM gauge symmetry do not necessarily satisfy this property [1][2][3]. Therefore, the apparent SM matter unification into the SU(5) multiplets is quite remarkable. In fact, the matter unification has been propelling the study of the grand unified theory (GUT) for a long time [4][5][6] (see Ref. [7] for reviews). In the conventional GUT, the SM matter fields are unified into common representations of a simple group containing SU (5). As a result of the matter unification, the GUT predicts the proton decay, which are extensively searched for in a variety of experiments (see e.g. Refs. [8][9][10][11]).
Recently, we have proposed a new framework "fake GUT" [12]. This framework can explain the apparent unification of the SM matter fields into the SU(5) multiplets in a different way than conventional GUT models. In the fake GUT, the SM matter fields are not necessarily embedded into common SU(5) multiplets at the high energy. Although the quarks and leptons can have different origins, they form complete SU(5) multiplets at the low energy as if they originate from the same multiplets. In the fake GUT, the prediction of the proton decay can be significantly different from that in conventional GUT models. As an extreme example, the quarks and leptons may reside in completely different multiplets. In such a case, the proton decay does not occur.
In Ref. [12], a fake GUT model based on a non-simple SU(5)×U(2) H group has been sketched. In this model, the leptons mainly reside in the part of the vector-like fermions of U(2) H , while the quarks reside in the 5 ⊕ 10 representations. After the spontaneous symmetry breaking of SU(5) × U(2) H down to the SM gauge group, only the leptons and the quarks remain massless which apparently form 5 ⊕ 10 representations. Although the "GUT scale" is predicted to be much lower than the conventional GUT scale, the proton decay rate is suppressed as the quarks and the leptons are not in the same multiplets. In this paper, we study details of the SU(5) × U(2) H model.
We also extend the non-simple SU(5) × U(2) H model to that based on a semi-simple group SU(5) × SU(3) H . With this extension, we can successfully explain the charge quantization and avoid the Landau pole problem of U(1) H gauge interaction. We also discuss the symmetry which determines the fraction of the leptons originating from 5 ⊕ 10, which in turn controls the proton decay rate.
The organization of this paper is as follows. In Sec. 2, we first review the idea of the fake GUT. In Sec. 3, we discuss details of the non-simple SU(5) × U(2) H group. In Sec. 4, we extend the model to that based on a semi-simple SU(5) × SU(3) H model. The final section is devoted to our conclusions.

Fake GUT
In this section, we review the idea of the fake GUT. Let us consider a high energy theory which satisfies the following conditions; 1. The gauge group is G = SU(5) × H, which is spontaneously broken down to the SM gauge group G SM = SU(3) c × SU(2) L × U(1) Y at the fake GUT scale.
2. Three copies of the chiral fermions in the 5 ⊕ 10 representations of SU (5) which are neutral under H.
3. Additional fermions which consist of the vector-like representations of G.
4. All the Cartan subgroups of SU (5) remain unbroken and all of them take part in G SM . 4 . Some of SU(3) c , SU(2) L and U(1) Y may be diagonal subgroups of SU(5) × H.
We call the theory which satisfies conditions 1 to 4 the "fake GUT." In this work, we also assume the additional condition 4 , which realizes more viable models in view of the proton decay and the coupling unification. As proven in Ref. [12], the fake GUT model guarantees that the low energy fermions completely match with the 5 ⊕ 10 representations, while the quarks and the leptons do not necessarily originate from 5⊕10 multiplets (see the App. A). When some of quarks/leptons originate from fields other than5 ⊕ 10, some of the 5 ⊕ 10 fermions become massive whose mass partners are parts of the vector-like fermions of G after G is spontaneously broken down to G SM . In this way, only the SM fermions remain massless at the GUT scale, which form the 5 ⊕ 10 representations regardless of how they are embedded in the fake GUT representations at the high energy [12]. This feature strongly owes to the nature of the chiral fermion and does not apply to the bosonic fields.
In the case of supersymmetry (SUSY), this argument applies to the chiral superfields. Let us emphasize that the apparent matter unification in the fake GUT is guaranteed by the matching of the characters of the chiral representations and does not depend on the details of the models. This argument is more versatile than the 't Hooft anomaly (i.e. the anomaly of the global symmetry) matching conditions to restrict the fermions in the low energy theory [13], since the latter relies on the global symmetries of the model.
We may consider the fake GUT with a larger group G UV than SU(5) × H, as long as G UV is broken to SU(5) × H satisfying the above conditions. For instance, SO(10) × H → SU(5) × H with the 16 representations satisfies the above conditions and works as a fake GUT model.
Another interesting feature of the fake GUT model is that it does not necessarily require the SM gauge coupling unification for a non-trivial H when it satisfies the condition 4 . Thus, although the coupling unification fails in the non-supersymmetric SM, the fake GUT framework works with a proper choice of H.
There are many possible choices of H. For example, H = 1, U(1) H , SU(N ) H · · · , are possible candidates. It is also possible to consider In fact, many extensions of the GUT models fit into the fake GUT framework. For instance, massive fermion extensions play important role in the Yukawa coupling unification [14,15], (see also Refs. [16][17][18] for recent applications), and in suppressing lepton/baryon number violation in GUT [19][20][21].
Not all possibilities are, however, phenomenologically viable due to the constraints from the gauge coupling matching conditions and the proton lifetime. As we will see later, we find that the minimal viable choice is H = U(2) H , if the running of the gauge coupling constants is the same as the SM below the fake GUT scale. In this case, the smaller choices such as H = 1, H = U(1) H or H = SU(2) H are not phenomenologically compatible. In the next section, we discuss a SU(5)×U(2) H model. We also extend the model so that U(2) H is embedded into SU(3) H in Sec. 4.

SU(5) × U(2) H model
In this section, we discuss a model with Below the fake GUT scale, SU(2) L × U(1) Y appear as the diagonal subgroups of SU(5) × U(2) H , while SU(3) c appears solely from SU(5). As we emphasized above, this choice is the minimal gauge group for the fake GUT model which is phenomenologically compatible. We summarize the matter contents of this model in Tab. 1. Note that the idea of the fake GUT is completely different from that of the SUSY GUT model based on the product group [22], although they are both based on the product gauge group. In the product group SUSY GUT model, the product group is used to solve the so-called the doublet- triplet mass splitting problem, while the effective gauge coupling unification is taken seriously and the matter multiplets are assumed as the GUT multiplets.

Origin of SM fermions
We introduce the three generations of the chiral multiplets 5 ⊕ 10 of SU (5). The SM right-handed down quarks d R sector fully come from the 5, and the right-handed up quarks u R and the left-handed SU(2) L doublet quarks q L from 10. 1 In addition, we introduce three pairs of vector-like multiplets charged under H = U(2) H , Here, the group representations are denoted by (SU(5), SU(2) H ) U(1) H . In this model, spontaneous breaking of SU(5)×U(2) H into G SM is achieved by a vacuum expectation value (VEV) of a complex scalar field φ 2 , which is a bi-fundamental representation, (5, 2) −1/2 . Explicitly, the VEV of φ 2 (see Ref. [23]) Here, λ L,E are coupling constants, and Λ cut a cutoff scale larger than the fake GUT scale. The summation of the flavor indices i, j = 1, 2, 3 is understood. The above interactions are the most general forms of fermion bilinears up to mass dimension five. The higher dimensional operator can be generated by integrating over the massive fermions in (5, 2) −1/2 representation coupling to φ 2 . Or, we may also consider a complex scalar φ 2 in (10, 1) 1 which has a Yukawa coupling E H φ 2 10 and a trilinear coupling φ 2 φ 2 φ 2 .
After the fake GUT symmetry breaking, the above interactions lead to the mass terms in the leptonic sector. The quarks are, on the other hand, fully contained in 5 ⊕ 10, and hence, remain massless. The mass terms of the leptonic sector are given by, where the mass matrices are given written as Here, 5 L and 10 E denote the components of the G SM gauge charges corresponding to those of the doublet and the singlet leptons. Since the mass matrices are given by 3 × 6 matrices, we find that three leptons remain massless due to the rank conditions. As a result, the three generations of the Weyl fermions of the SM can be realized, although the leptons do not fully belong to 5 ⊕ 10.
To see the origin of the massless leptons, let us omit the flavor mixing and focus on a single generation. In this case, the mass eigenstates are given by where L and e R are the doublet and the singlet massless eigenstates, respectively. The mixing angles, θ L,E , are where we have taken the parameters real positive. The masses of the heavy leptons are given by, As an extreme example, the SM leptons are fully contained in L H and E H for m L = m E = 0, while the heavy leptons remain massive. In this case, although the quarks and the leptons in the SM originate from completely separate multiplets in the fake GUT, the quarks and the leptons apparently form 5 ⊕ 10 multiplets at the low energy. Note that the limit m L = m E = 0 enhances a global symmetry under which L H and E H are charged (see Sec. 3.4).

Origin of SM Higgs and Yukawa interactions
In the previous subsection, we discussed the origin of the SM Weyl fermions. Here, we discuss the origin of the SM Higgs boson and the Yukawa interactions. As the quarks and the leptons can originate from the separate multiplets, the SM Yukawa interactions consist of various contributions. Here, we consider a case that only one SM Higgs doublet remains in the low energy.
Concretely, we introduce H 5 and H 2 scalar fields, which are of (5, 1) 0 and (1, 2) 1/2 representations. These scalar fields contain the SM Higgs scalars as well as the colored Higgs, (3.12) The SM Higgs components in H 5 and H 2 mix with each other, via an interaction, once φ 2 develops the VEV. As a result, the effective mass terms of H 5 and H 2 are given by, Accordingly, the SM Higgs, h SM , is given by a linear combination of h SM 2 and h SM 5 , Here, m 2 5 , m 2 5 , and m 2 2 are orders of O(v 2 2 ) and include the contribution from the VEV of φ 2 . We also assume µ mix is of O(v 2 ). To achieve the mass of h SM in O(100) GeV, we require severe fine-tuning as in the case of the conventional GUT.
By using H 5 and H 2 , the origins of the SM Yukawa interactions are given by, where i and j run the number of generations. The SM Yukawa couplings are obtained by substituting → cos θ h h SM after diagonalizing the mass matrices in Eq. (3.6).
As we will see in Sec. 3.5, the lepton components in 5 ⊕ 10 should be highly suppressed to evade the constrains from the proton decay, i.e., θ L,E 1. In this case, the SM Yukawa couplings are approximately given by,

Gauge coupling constants
In the present setup, SU(2) L ×U(1) Y gauge symmetries are the diagonal subgroups of SU(5)×U(2) H , and SU(3) c gauge symmetry is the remaining unbroken subgroup of SU (5). The tree-level matching conditions of the gauge coupling constants of the SM and the SU(5) × U(2) H model at the fake GUT scale are given by [24], where α i = g 2 i /(4π), and g i 's are the gauge coupling constants. Here, M X is the mass of X µ in Tab. 2, and we assume that all the supermassive particles have the masses of the same order of the magnitude. The fine structure constants α 5 , α 2H and α 1H are those of SU(5) and SU(2) H × U(1) H , respectively. 2 Thus, the gauge couplings in the SM do not unify at the fake GUT scale.
As an interesting feature of SU(5) × U(2) H model, there is an upper-limit on the fake GUT scale. From the matching conditions in Eq. (3.19), the fake GUT scale is consistent only for α −1 Fig. 1, we find the fake GUT scale is lower than M upper Here, let us comment on the coupling matching condition for other choice of the gauge group H. For the model with SU for example, the matching condition of the gauge couplings similar to Eq. (3.19), requires that α −1 1,3 > α −1 2 . This is not possible unless G SM charged particles other than the SM fields appear below the fake GUT scale (see Fig. 1). In this way, the matching condition and the gauge coupling running of the SM constraints the choice of H.

Global lepton and baryon symmetries
The present model possesses the global U(1) 5 symmetry (fiveness) as in the conventional GUT model. The charge assignment of the fiveness enlarged to the U(2) H sector is given in Tab. 3. Note that the interaction terms in Eqs.
where Q 5 and Q Y are the charges of U(1) 5 and U(1) Y . The B − L applies to the fermion fields in the same way of the SM. As mentioned earlier, an additional global symmetry is enhanced in the limit of m L,E → 0, which we call U(1) LH symmetry. The charge assignment is given in Tab. 3. The U(1) LH symmetry also remains in the low energy which gives an additional lepton symmetry, U(1) L . This lepton symmetry is, however, not exact since it is broken by the gauge/gravitational anomalies. This additional symmetry can be identified with the lepton symmetry of the SM.
Up to the gauge/gravitational anomalies the U(1) B−L and U(1) L symmetries are conserved separately in the limit of m L,E → 0, which may be rearranged as U(1) B and U(1) L . As far as these symmetries are respected, no visible low energy baryon/lepton violating processes are expected. Note that the effects of the breaking of U(1) B−L and U(1) L by the gauge and the gravitational anomalies are highly suppressed for the nucleus decays [25].
In the conventional SU(5) GUT both the baryon symmetry U(1) B and the lepton symmetry U(1) L are embedded in the GUT fiveness. Thus, at the low energy, only the linear combination, U(1) B−L , is conserved. As a result, the conventional GUT predicts the proton decay which violates the U(1) B and U(1) L symmetries while U(1) B−L is conserved. The proton lifetime is dominantly determined by the GUT gauge boson mass scale.
In the present fake GUT model, the proton decay is forbidden when the U(1) LH symmetry emerges, i.e. m L,E = 0. In fact, this symmetry is crucial for the successful U(2) H fake GUT model as the X gauge boson mass scale M X < 10 14.4 GeV, which would lead to too rapid proton decay without this symmetry.
It is, however, highly non-trivial whether we can impose such global symmetries on the model. In general, it is argued that all global symmetries are broken by quantum gravity effects (see e.g. [26][27][28][29][30][31][32]). Moreover, the gauge/gravitational anomalies already break the U(1) B−L and U(1) L symmetries explicitly in this model. In addition, the origins of the neutrino masses and the baryon asymmetry in the Universe may also indicate the breaking of those symmetries. Therefore, we expect that small violation of U(1) B−L and U(1) L exist which generate tiny m L,E . The breaking of the U(1) B−L and U(1) L symmetries depend on the further high energy physics. In the following analysis, we simply regard m L,E (or equivalently θ L,E in Eqs. (3.9) and (3.10)) as the parameter of the explicit symmetry breaking. 3 Let us also comment on the origin of the neutrino mass. The observations of the neutrino oscillations show that the neutrinos have tiny masses. In the present model, there are various ways to realize the active neutrino masses. First, we may consider the active neutrinos as the Dirac neutrinos, where the right-handed neutrinos couple to the leptons and Higgs in the U(2) H sector. In this case, there is no lepton symmetry violation, and hence, the proton decay is also suppressed for m L,E → 0 as explained above.
Another possibility is to couple the right-handed neutrinos to the SU(5) sector. In this case, although the active neutrino masses are the Dirac type, we require the lepton symmetry breaking, that is, the tiny mixing between the lepton components of 5 and L H . It is an interesting feature of this model, that both the neutrino masses and the proton decay rate are generated by the effects of the lepton symmetry breaking, i.e., m L .
More attractive possibility is to consider the Majorana neutrino masses, which break U(1) LH symmetry down to Z 2 subgroup. When Z 2 subgroup of U(1) LH is conserved separately from U(1) B−L , the proton is stable as long as Z 2 is unbroken. For instance, it is possible to consider the seesaw mechanism [33][34][35][36][37] in the U(2) H sector where the model possesses the Z 2 symmetry while m L,E are suppressed.

Nucleon decay
The nucleons can decay through the heavy SU(5) gauge bosons X exchange. For simplicity, we assume θ E,L in Eq. (3.7) and (3.8) do not depend on the generations. In this case, the proton lifetime of the p → π 0 + e + mode is (3.21) In order to estimate the proton lifetime, we adopt the method described in Ref. [38], using the hadron matrix elements of Refs. [39,40]. By comparing to the current limit, τ (p → π 0 + e + ) > 2.4 × 10 34 yrs [8], this model seemingly predicts too short proton lifetime as in the case of the conventional non-supersymmetric SU(5) GUT model. However, the proton decay width can be suppressed by the mixing angles θ E,L in the fake GUT. In the above example, small mixings, sin θ E,L 10 −4 , are consistent with the current limit on the proton lifetime. This suppression means that the SM quarks and leptons mostly have different origins. Such small mixing angles correspond to m E,L v 2 , where the global symmetry is enhanced, as discussed in the previous subsection.
The proton lifetime in Eq. (3.21) shows that we need small mixings of both the 5 L and 10 E with the SM leptons to evade the constraints. Thus, for the choices such as H = 1, U(1) H , SU(2) H , for example, the proton lifetime is predicted to be too short unless we introduce a large number of SU(5) charged fermions [20]. This is the reason why SU(5) × U(2) H is the minimal choice which is phenomenologically viable.
Note that we estimate the above proton lifetime without considering the effects of the flavor mixings for simplicity. In general, however, the lepton mass matrices in Eq. (3.6) are flavor dependent. Hence, an SU(5) gauge interaction eigenstate does not coincide with one generation of the SM fermions but consists of the admixture of the multiple SM generations. Therefore, the predictions of the nucleon decay rates and the branching fractions in the fake GUT are different from those in the conventional GUT. For example, the decay rate of the p → π 0 + µ + mode can be larger than that of the p → π 0 + e + mode. This is a contrary to the conventional GUT, where the nucleon decay modes which include different generations are suppressed by the Cabibbo-Kobayashi-Maskawa (CKM) mixing angle. In this way, a variety of the nucleon decay modes will provide striking signatures of the fake GUT.
To see the effects of the flavor dependence of the mass matrices in Eq. (3.6), let us consider the following example of the fermion mass terms at the fake GUT scale. Here, we introduce small parameters δ L,E,ij with |δ L,E,ij | 1, which represents explicit breaking of the global symmetry. We take flavor indices of 5 i and 10 i to match the generations of the quarks, and those of L Hi and E Hi approximately corresponds to the generations of the leptons (see Eq. (3.18)).
In our analysis, we adopt a flavor basis that the up-type Yukawa matrix is diagonal, while the down-type Yukawa matrix has CKM-related off diagonal elements.
In Fig. 2, we show the proton lifetime for each mode. In each figure, we switch on the mixing parameters in Eq. (3.22) as shown in the caption. The predictions are proportional to δ −2 L,E and M 4 X . Black lines denote the Super-Kamiokande constraints at 90% C.L. [8,[41][42][43][44][45] and yellow ones represent the future prospects of Hyper-Kamiokande [9]. The figures show the model predicts various leading proton decay modes depending on the mixing parameters. For the case of the Fig. 2 (b), for example, p → π 0 + µ + mode is the leading decay mode, which is not expected in the conventional GUT model.
Finally, let us comment on the proton decay through the colored Higgs exchanges. In the conventional GUT model, these contributions is always subdominant compared with those of the X boson exchanges due to the Yukawa suppression [46], Here, we multiply the factor sin 4 θ h which stems from the relative enhancement of y 5,10 , compared with the SM Yukawa couplings given in Eq. (3.18). To reproduce the masses of the SM fermions, we find that, the largest components of y 10,5,LE are Thus, by requiring |y 10,5,LE | 1, sin θ h ∼ y t /y 10 0.5 , cos θ h ∼ y τ /y LE 10 −2 .
In the previous section, we showed that SU (5)

Spontaneous symmetry breaking of SU(5) × SU(3) H
In this model, we introduce a real scalar field, A, in (1, 8) representation, and a complex scalar field, φ 3 , in (5, 3) representation of (SU(5), SU(3) H ), respectively. The scalar potential is  (SU (5) Here, µ's are parameters with mass dimension one, and λ's are dimensionless parameters. We take all the parameters are positive for simplicity. The vacuum in SU(5) × U(2) H model in Eq. (3.3) is reproduced when A and φ 3 take the VEVs in the following form; To achieve this form, v A = 0 is crucial, since otherwise we find that either only a single row of φ 3 or all the three rows of φ 3 are non-vanishing. 4 Once v A = 0, the last two terms in Eq.(4.2) can make the mass squared of the third row of φ 3 positive while those of the first and the second rows are kept negative. For v A v 3 , we may regard that the symmetry breaking takes place in the two steps as follows; where the second step corresponds to the symmetry breaking in the SU(5)×U(2) H model. Note that the model does not require the hierarchy between v A and v 3 , and it is possible to consider situation where SU(5) × SU(3) H breaks down directly to G SM for v A < v 3 .
In Tab. 5, we show the gauge boson masses. The X boson absorbs (3, 2) −5/6 scalar in Tab. 4 as a longitudinal mode. The Ω 2 boson absorbs a linear combination of two (1, 2) +3/2 scalar fields. The Ω 3 (Ω 1 ) absorb a linear combination of (1, 3) 0 ((1, 1) 0 ) in the bi-fundamental fake GUT Higgs φ 3 . Accordingly, the physical fake GUT Higgs bosons appearing from φ 3 and A consist of a (1, 2) +3/2 scalar field, two (1, 3) R 0 scalars, two (1, 1) R 0 scalars, and a (3, 1) +2/3 scalar. In the previous SU(5)×U(2) H model, the matching conditions between the SM and SU(5)×U(2) H gauge couplings are given in Eq. (3.19). For v A v 3 the matching conditions of the gauge coupling constants at the symmetry breaking in the second step in Eq. (4.4) are also given by Eq. (3.19) with M X replaced by M X = g 5 v 3 / √ 2. The matching conditions between SU(5) × U(2) H and SU(5) × SU(3) H models are, on the other hand, given by Here, M Ω = 3g 3H v 2 A + v 2 3 /18 is the energy scale at which these two models are matched. The factor 1/3 in Eq. (4.5) appears since the U(1) H charge is embedded in the fundamental representation of SU(3) H with the normalization, tr[t a t b ] = δ ab /2, with t a being the generator of SU(3) H . 5 For a given M X , the gauge couplings α 1H and α 2H are determined by Eq. (3.19). Above M X , α 5 , α 1H , and α 2H evolves following the renormalization group (RG) equations in the SU (5)  Since M Ω M X in the present model, we find that there is an upper limit on M X which can be   (see Eq. (3.21)). To achieve the highly suppressed lepton mixing angle, we need a high quality lepton symmetry. In the App. C, we give an example of a model in which such a high quality lepton symmetry originates from a discrete gauge symmetry.

SM fermions, SM Higgs and Yukawa interactions
Here, m T is a mass parameter of the vector-like fermions and λ's are dimensionless couplings. The flavor indices are omitted for simplicity. The SU(3) H indices of L T φ † 3 φ † 3 are contracted by the totally anti-symmetric invariant tensor of SU(3) H , while its SU(5) indices are contracted with those of 10 which is the anti-symmetric representation. The cutoff scale Λ cut is a scale larger than the fake GUT scale v 3 and v A corresponding to some heavier particles. An example of such particles is Dirac fermion with (5,3). We may also consider a complex scalar φ 3 in (10, 3) which has a Yukawa coupling L T φ 3 10 and a trilinear coupling φ 3 φ 3 φ 3 (see the App. C for a concrete model.).
As in the previous section, the SM quarks originate from 5 ⊕ 10. The lepton components in 5 and 10 and L T 's obtain 3 × 6 mass matrices from the interactions in Eq. (4.9), and the three leptons remain massless due to the rank conditions. Since the 5 ⊕ 10 contributions to the SM leptons are highly suppressed (see Eq. (4.7)), we require m T v 3,A and λ T AT 1. As in the case of the SU(5) × U(2) H model, the global U(1) LT symmetry is enhanced in addition to the global U(1) 5 symmetry in the limit of vanishing m T and λ T AT . In the App. C, we discuss a model where the lepton symmetry originates from a discrete gauge symmetry.
Next, we discuss the origin of the SM Higgs and Yukawa interactions in the SU(5) × SU(3) H model. As in the SU(5)×U(2) H model, the SM Yukawa interactions come from various contributions, and we consider a case that only one SU(2) L Higgs doublet remains in the low energy. We introduce H 5 and H 3 scalar fields, which are (5, 1) and (1, 3) which is the naive extension of Eq. (3.17). Here, the subscript α is the index of the SU(3) H and i and j run the number of generations. Since αβγ is totally antisymmetric, we find (y LT ) ij = −(y LT ) ji . The 3-by-3 antisymmetric Yukawa coupling results in the massless electron and the µ and τ lepton in the same mass. Thus, the Yukawa coupling given by Eq. (4.12) does not reproduce the lepton masses in the SM. As in Eq. (3.18), the lepton Yukawa couplings receive the contribution from y 5 , which are suppressed by the lepton mixing angles θ L × θ E . The lepton mixing angles are, however, required to be highly suppressed, θ L,E 10 −12 , to evade the constraints from the proton lifetime (see Eq. (4.7)). Thus, those contributions are too small to reproduce the lepton masses in the SM, and hence, we need other origins of the lepton Yukawa interactions.
As an example to generate appropriate lepton Yukawa coupling, we consider the following higher dimensional operator, where Λ Y is a cutoff scale. The scale Λ Y can be given by the mass of heavy particles such as Dirac fermions in (1, 3) representation (see App. C). Once A obtains the VEV, this operator generates the lepton Yukawa couplings. Unlike the Yukawa couplings in Eq. (4.12), the 3-by-3 coefficient matrix is not anti-symmetric, and hence, this operator can reproduce the SM lepton spectrum, when Λ Y is smaller than O(10 2 ) × v A .

Conclusions and discussions
The fake GUT is a framework which has been proposed to explain the perfect fit of the SM matter fields into the SU(5) multiplets [12]. Unlike the conventional GUT [4], the quarks and leptons are not necessarily embedded in common GUT multiplets but embedded in different multiplets although they appear to form complete GUT multiplets at the low energy. In this paper, we studied details of the model based on SU(5) × U(2) H gauge symmetry and its extension with SU(5) × SU(3) H gauge symmetry. We discussed the nature of the fake GUT symmetry breaking. We also studied the origins of the SM quarks/leptons, the Higgs fields, and the SM Yukawa interactions.
In the present models, the global lepton and baryon symmetries play a crucial role to avoid too rapid nucleon decays. However, these global symmetries are less likely exact ones, due to theoretical and phenomenological reasons. With the violation of the global symmetries, the nucleons are no longer stable. The decay rates and decay modes strongly depend on its size and flavor structure. Observations of multiple nucleon decay modes are the smoking guns of the present models.
We extended the SU (5) (7), respectively. However, the resultant SU(5) × U(2) H models do not satisfy the conditions of the fake GUT model, and hence, they do not provide viable UV completions of the SU(5) × U(2) H model.
Finally let us comment on the topological objects generated in the early Universe in these models. Unlike the conventional GUT based on simple groups, the SU(5) × U(2) H model has no dangerous topological objects. Therefore high-scale inflation and high reheating temperature are allowed in the model. In the case of SU(5) × SU(3) H , on the other hand, it is possible to generate monopoles if the fake GUT breaking takes place after the inflation. In order to avoid this monopole problem, the reheating temperature should be much less than the fake GUT scale. Detailed constraints and the possible detection of the monopole will be explored elsewhere. Besides, let us also comment on the monopole catalysis of proton decay. In the conventional GUT model, the monopoles induce the baryon-number non-conserving reactions [47,48]. In the fake GUT model, on the other hand, baryon-number violating processes, e.g. proton decay, are also controlled by the lepton symmetry unlike the conventional GUT. This may provide different features of the monopole in the fake GUT model compared to that in the conventional GUT.

A Characters of SM and SU(5) groups
In order to see that the SM fermions are apparently unified into the SU(5) multiplets, it is convenient to use the character of the gauge groups G defined by the trace of a representation matrix, R(g) (g ∈ G) (A.1) In this paper, we use the left-handed Weyl fermions, and the representation matrices are defined for those left-handed Weyl fermions. In this definition, the chiral nature of the representations of the fermions is encoded in the following quantity, to which only the chiral fermions give non-zero contributions [49]. Now, let us consider ∆χ R in the SM model. Since the SM model fermions are the chiral fermions, all of them contributes to ∆χ R which amounts to where n gen = 3 is the number of the fermion generations. As a surprising feature of the SM fermions, A SM (g SM ) coincides with the ∆χ R of the 5 ⊕ 10 representation of SU (5), that is where g SM is the SU(5) elements restricted to the SM gauge group. By remembering the orthogonality and the completeness of the characters, Eq. (A.4) means that the SM fermions can be exactly embedded into the three copies of 5 ⊕ 10. This amazing feature cannot be explained within the SM, because, in general, chiral fermions consistent with SM gauge symmetry do not necessarily satisfy this property.
The fake GUT conditions in Sec. 2 guarantee the relation in Eq. (A.4) automatically. Let us emphasize again that the quarks and leptons are not required to reside in the same SU(5) multiplets in the fake GUT model unlike the conventional GUT model. Once Eq. (A.4) is guaranteed by the fake GUT model, it uniquely determines the gauge charges of the low energy fermions under G SM due to the orthogonality and the completeness of the characters. In this way, the fake GUT model explains why the SM fermions form the apparently complete SU(5) multiplets.

B RG Equations
We show the RG equations of the gauge couplings of the SU(5) × U(2) H model and SU(5) × SU(3) H model up to two-loop level. We used the program PyR@TE 3 [50] for the calculation. Here, we neglect the Yukawa and Higgs couplings.
The matter fields are given in Tab. 1.
The matter fields are given in Tab. 4.

C On global lepton and baryon symmetry
As discussed in subsection 3.4, the global lepton symmetry plays a crucial role to make the SU(5) × U(2) H model phenomenologically viable. The global lepton symmetry is more important for SU(5)× SU(3) H model due to the smaller X gauge boson mass. In this appendix, we give an example of the model in which the high quality lepton symmetry originates from a discrete gauge symmetry in SU(5) × SU(3) H model. Before discussing the lepton symmetry, we first provide a concrete model to generate the higher dimensional operators used in Eqs. (4.9) and (4.13). As for the mixing term in Eq. (4.9), we consider a complex scalar φ 3 in (10, 3) representation which has a Yukawa coupling L T φ 3 10 trilinear coupling φ 3 φ 3 φ 3 . When the mass of φ 3 is larger than the fake GUT scale while the coefficient of the trilinear coupling is of order of the fake GUT scale, the VEV of φ 3 is aligned to the φ 3 ∼ φ † 3 φ † 3 . In this way, the higher dimensional lepton mixing mass is achieved by Here, M φ and µ φφ φ are the mass and the coefficient of the trilinear coupling of φ , respectively.
To generate the origin of the Yukawa interactions in Eq. (4.13), we introduce other Dirac fermions, (H F , H F ) with the Yukawa interactions, We again assume that (H F ,H F ) is heavier than the fake GUT scale. By integrating out H F , H F , we obtain, Since we have the other Yukawa coupling in Eq. (4.12), one pair of (H F , H F ) can reproduce the masses of the SM leptons with appropriate choice of a i , b i and (Y LT ) ij . In Sec. 3.4 we assumed a global lepton symmetry to suppress the proton decay rate. In the SU(5) × SU(3) H model, the lepton symmetry is required to suppress m T and λ T . In the present model, however, the lepton Yukawa interactions in Eq. (4.12) and (4.13) include L 2 T , and hence, the lepton symmetry can not be the continuous U(1) symmetry. Instead, we consider a discrete Z 2n (n ∈ N) symmetry, under which L T 's have charge n. The discrete Z 2n symmetry forbids only the unwanted m T and λ T terms when (H F , H F ) also have Z 2n charge n while other fields are neutral under Z 2n (see Tab. 6).
As mentioned earlier, it is argued that all global symmetries are broken by quantum gravity effects (see e.g., Refs. [26][27][28][29][30][31][32]). Thus, we seek a possibility to realize Z 2n symmetry. The anomaly coefficients of the gauged discrete symmetry come only from L T (n) since (H F , H F ) contributions trivially cancels; which should be vanishing for the Z 2n symmetry to be a gauge symmetry [51][52][53][54]. The third condition in Eq. C.6 can be always satisfied by choosing the normalization of n appropriately [55]. Thus, it does not lead to useful constraints on the particle contents. Here, the first, second and third quantities in the each multiplication represent Z 2n charge, the number of particles with same charges and degrees of freedom of SU(3) H . Thus, to realize the gauged Z 2n symmetry, we need additional SU(3) H charged fermions which are chiral under Z 2n . Table 6: Field contents of fermions and a scalar with the Z 2n charge and pairing fermions for n = 4. SM lepton doublets are in L T 's. The Z 8 symmetry is free from anomaly including Z 3 8 . In practice, this Z 8 symmetry is equivalent to Z 4 symmetry.

SU(5) SU(3)
To cancel the anomaly, Z 2n × [SU(3) H ] 2 , we introduce a pair of (Θ, Θ) which are (anti)fundamental representation of SU(3) H and have Z 2n charges (n,0). In this case, all the anomaly coefficients vanish, and Z 2n can be the gauge symmetry. However, Z 2n must be broken to give a mass to the pair, (Θ, Θ), by the VEV of a complex scalar φ Z with the Z 2n charge n. Such a VEV of φ Z also generates m T , which results in the large lepton mixing angles.
To avoid this problem, we assume that the additional chiral fermions have smaller Z 2n charges. Concretely, we consider n = 4, and hence, Z 8 symmetry, with the Z 8 charges of (Θ, Θ) being (2,0) (see Tab. 6). In this case, the anomaly coefficients of Z 8 × [SU(3) H ] 2 and Z 8 × [gravity] 2 require two pairs of (Θ, Θ). In this way, we achieve the anomaly-free Z 8 symmetry.
The masses of the pairs of (Θ, Θ) are given by the VEV of the complex field φ Z with a Z 8 charge 6, v Z = φ Z . The mass of a pair (H F , H F ) is not forbidden by the Z 8 symmetry. 6 The lepton symmetry breaking parameters m T and λ T AT are, on the other hand, suppressed by the cutoff scale which is now taken to be the Planck scale, M Pl , To achieve those mass parameters smaller than 10 −4 GeV, we find that the breaking scale of Z 8 is limited from above, v Z < O 10 7 GeV . (C. 8) In summary, the extra particles (Θ, Θ) have masses of v Z 10 7 GeV, while the lepton symmetry breaking mass parameter m T is highly suppressed.
When a discrete symmetry is broken spontaneously, the domain walls are formed. To avoid the cosmological domain wall problem, we need to assume that at least the reheating temperature after inflation should be lower than the Z 8 breaking scale, v Z . Combined with the upper limit on v Z in Eq. (C.8), the upper limit on the reheating temperature is given by T rh O(10 7 ) GeV.
The running of the gauge coupling of the SU(3) H is modified because of the addition of multiple Dirac fermions and scalar particles in this example model. However, the asymptotic freedom SU(3) H gauge interaction is still preserved in the present setup.