Tri-unification: a separate SU (5) for each fermion family

: In this paper we discuss SU (5) 3 with cyclic symmetry as a possible grand unified theory (GUT). The basic idea of such a tri-unification is that there is a separate SU (5) for each fermion family, with the light Higgs doublet(s) arising from the third family SU (5), providing a basis for charged fermion mass hierarchies. SU (5) 3 tri-unification reconciles the idea of gauge non-universality with the idea of gauge coupling unification, opening the possibility to build consistent non-universal descriptions of Nature that are valid all the way up to the scale of grand unification. As a concrete example, we propose a grand unified embedding of the tri-hypercharge model U (1) 3 Y based on an SU (5) 3 framework with cyclic symmetry. We discuss a minimal tri-hypercharge example which can account for all the quark and lepton (including neutrino) masses and mixing parameters. We show that it is possible to unify the gauge couplings into a single gauge coupling associated with the cyclic SU (5) 3 gauge group, by assuming minimal multiplet splitting, together with a set of relatively light colour octet scalars. We also study proton decay in this example, and present the predictions for the proton lifetime in the dominant e + π 0 channel.


Introduction
The flavour problem remains one of the most intriguing puzzles of the Standard Model (SM), being responsible for most of its parameters.The origin of three families, which are identical under the SM gauge group, but differ greatly in mass, with the quark mixing being small while the lepton mixing is large, is not addressed, while the origin of CP-violation only adds to the mystery.It is quite common to address these puzzles by assuming that the fermions are distinguished by a new spontaneously broken family symmetry, however this is not the only way forwards.
Recently two of us proposed an embedding of the SM based on the existence of one local weak hypercharge associated to each fermion family [1], where each SM fermion family i = 1, 2, 3 is charged only under their corresponding U (1) Y i factor.Such a framework avoids the family replication of the SM and is naturally anomaly-free.If the Higgs doublet(s) only carry third family hypercharge, then the third family is naturally heavier and the light families are massless in first approximation, providing a novel way of addressing the flavour problem.
In this paper we propose a grand unified embedding of the tri-hypercharge [1] model based on a non-supersymmetric SU (5) 3 framework with cyclic symmetry.This is a generalisation of SU (5) grand unification [2] in which we assign a separate SU (5) group to each fermion family, together with a cyclic symmetry to ensure gauge coupling unification 1 .We discuss a minimal example which can account for all the quark and lepton (including neutrino) masses and mixing parameters.We show that it is possible to unify the gauge couplings into a single gauge coupling associated with the cyclic SU (5) 3 gauge group, by assuming minimalistic multiplet splitting, together with a set of relatively light colour octet scalars.We also study proton decay in this example, and present the predictions of the proton lifetime in the dominant e + π 0 channel.More generally, the SU (5) 3 framework proposed here may embed a broader class of gauge nonuniversal models, reconciling the ideas of gauge non-universality with gauge coupling unification at the GUT scale.In particular, SU (5) 3 may embed theories based on the family decomposition of the SM group, such as the SU (2) 3  L model [10][11][12][13][14][15], the SU (3) 3 c model [16] or the aforementioned tri-hypercharge model, as an alternative to the existing ultraviolet (UV) completions which are all based on (variations of) the Pati-Salam (PS) group [17][18][19][20][21][22][23][24].We note however that while most of the previous papers explain the origin of the flavour structure of the SM, none of them provides a gauge unified framework.In general, gauge non-universal models can address the flavour puzzle at the price of complicating the gauge sector, which in such theories may contain up to nine arbitrary gauge couplings in the UV.
The layout of the remainder of the paper is as follows.In section 2 we discuss a general SU (5) 3 framework for model building.In rather lengthy section 3 we analyse an example SU (5) 3  unification model breaking to tri-hypercharge, including the charged fermion mass hierarchies and quark mixing, neutrino masses and mixing, gauge coupling unification and proton decay.Section 4 concludes the paper.In Appendix A we detail the energy regimes, symmetries and particle content Field SU (5) 1 SU (5) 2 SU (5) 3 of the considered example.In Appendix B we tabulate all possible hyperon embeddings in SU (5) 3representations with dimension up to 45.

General SU (5) 3 framework for model building
The basic idea is to embed the SM gauge group into a semi-simple gauge group containing three SU (5) factors, SU (5) 1 × SU (5) 2 × SU (5) 3 , ( where each SU (5) factor is associated to one family of chiral fermions i = 1, 2, 3.Moreover, we incorporate a cyclic permutation symmetry Z 3 that relates the three SU (5) factors, in the spirit of the trinification model [25].This implies that at the high energy GUT scale where SU (5) 3 is broken (typically in excess of 10 16 GeV) the gauge couplings of the three SU (5) factors are equal by cyclic symmetry, such that the gauge sector is fundamentally described by one gauge coupling.Therefore, although SU (5) 3 is a not a simple group, it may be regarded as a unified gauge theory.The motivation for considering such an SU (5) 3 with cyclic symmetry is that it allows gauge non-universal theories of flavour to emerge at low energies2 from a gauge universal theory, depending on the symmetry breaking chain.In the first step, SU (5) 3 may be 3 broken to three copies of the SM gauge group SM 3 .Then at lower energies, SM 3 is broken to some universal piece G universal consisting of some diagonal subgroups, together with some remaining family groups G 1 × G 2 × G 3 .If the Higgs doublet(s) transform non-trivially under the third family group G 3 , but not under the first nor second, then third family fermions get natural masses at the electroweak scale, while first and second family fermions are massless in first approximation.Their small masses naturally arise from the breaking of the non-universal gauge group down to the SM, which is the diagonal subgroup, and an approximate U (2) 5 flavour symmetry emerges, which is known to provide an efficient suppression of the most dangerous flavour-violating effects for new physics [28,29].
At still lower energies, the non-diagonal group factors G 1 × G 2 × G 3 are broken down to their diagonal subgroup, eventually leading to a flavour universal SM gauge group factor.This may happen in stages.It has been shown that the symmetry breaking pattern naturally explains the origin of fermion mass hierarchies and the smallness of quark mixing, while anarchic neutrino mixing may be incorporated via exotic variations of the type I seesaw mechanism [1,20].
Minimal examples of this class of theories include the tri-hypercharge model [1], which we shall focus on in this paper, where the universal (diagonal) group consists of the non-Abelian SM gauge group factors G universal = SU (3) c × SU (2) L while the remaining groups are the three gauge weak hypercharge factors L model [10][11][12][13][14][15], where There also exists the SU (3) 3 c model [16] (which is only able to explain the smallness of quark mixing), where Variations of these theories have been proposed in recent years, several of them assuming a possible embedding into (variations of) a Pati-Salam setup [17][18][19][20][21][22][23][24]30].
All these theories share a common feature: they explain the origin of the flavour structure of the SM at the price of complicating the gauge sector, which may now contain up to nine arbitrary gauge couplings.We will motivate that SU (5) 3 as the embedding of general theories G universal × G 1 × G 2 × G 3 resolves this issue, by unifying the complicated gauge sector of these theories into a single gauge coupling.The main ingredients of our general setup are as follows: • The presence of the Z 3 symmetry, which is of fundamental importance to achieve gauge unification, imposes that the matter content of the model shall be invariant under cyclic permutations of the three SU (5) factors.This enforces that each SU (5) factor contains the same representations of fermions and scalars, i.e. if the representation (A, B, C) is included, then (C, A, B) and (B, C, A) must be included too.
• Each family of chiral fermions i is embedded in the usual way into 5 and 10 representations of their corresponding SU (5) i factor, that we denote as 1.This choice is naturally consistent with the Z 3 symmetry.
• In a similar manner, three Higgs doublets H 1 , H 2 and H 3 are embedded into 5 representations, one for each SU (5) i factor.Notice that in non-universal theories of flavour it is commonly assumed the existence of only one Higgs doublet H 3 , which transforms only under the third site in order to explain the heaviness of the third family.This way, the SU (5) 3 framework involves the restriction of having three Higgses rather than only one, but we will argue that if the Z 3 symmetry is broken below the GUT scale, then only the third family Higgs H 3 may be light and perform electroweak symmetry breaking, while H 1 and H 2 are heavier and may play the role of heavy messengers for the effective Yukawa couplings of the light families.
• Higgs scalars in bi-representations connecting the different sites may be needed to generate the SM flavour structure at the level of the L or (Y, −Y ) scalars in tri-hypercharge (the so-called hyperons).These can be embedded in the associated bi-representations of SU (5) 3 , e.g.(5,5) scalars, (10, 10) scalars and so on.In Appendix B we tabulate all such scalars from SU (5) 3 representations with dimension up to 45, along with the hyperons that they generate at low energies.
• Finally, three scalar fields in bi-adjoint representations of each SU (5), Ω ij , spontaneously break the tri-unification symmetry.The three Ω ij are enough to perform both horizontal and vertical breaking of the three SU (5) groups at the GUT scale, down to the non-universal gauge group G universal × G 1 × G 2 × G 3 of choice that later explains the flavour structure of the SM (e.g.tri-hypercharge or SU (2) 3 L ).Another possibility that we will explore is breaking SU (5) 3 first to three copies of the SM (one for each family) and then to To summarise, the general pattern of symmetry breaking we assume is as follows 4 , where the SM 3 step is optional but may be convenient to achieve unification.In particular, the first step of symmetry breaking makes use of three SM singlets contained in Ω ij , while the second step may be performed via the remaining degrees of freedom in Ω ij , depending on the details of the low energy gauge theory that survives.The two final breaking steps are performed by Higgs scalars connecting the different sites that need to be specified for each particular model.
Beyond the general considerations listed in this section, when building a specific model one needs to choose the symmetry group G universal × G 1 × G 2 × G 3 , and add explicit scalars and/or fermion messengers that mediate the effective Yukawa couplings of light fermions.
Finally, one needs to study the Renormalization Group Equations (RGEs) of the various gauge couplings at the different steps all the way up to the SU (5) 3 scale where all gauge couplings need to unify.This is not a simple task, but we shall see that the relatively light messengers required to generate the effective Yukawa couplings, along with the presence of the approximate Z 3 symmetry at low energies, may naturally help to achieve unification.In the following, we shall illustrate this by describing a working example of the SU (5) 3 framework based on tri-hypercharge [1], where the various gauge couplings of the tri-hypercharge model unify at the GUT scale into a single gauge coupling.

An example SU (5) unification model breaking to tri-hypercharge
We now turn to the main example of interest, namely In this example, the basic idea is that SU (5) 3 breaks, via a sequence of scales, to the low energy (well below the GUT scale) tri-hypercharge gauge group with a separate gauged weak hypercharge for each fermion family, In [1] it was shown that the low energy tri-hypercharge model can naturally generate the flavour structure of the SM if spontaneously broken to SM hypercharge in a convenient way.The minimal setup involves the vacuum expectation values (VEVs) of the new Higgs "hyperons" At the GUT scale, the hyperons are embedded into bi-5 and bi-10 representations of SU (5) 3 expressed as Φ T,F ij , which must preserve the cyclic symmetry, as shown in Table 2.Although this involves the appearance of many hyperons (and other scalars) beyond the minimal set of hyperons that we need, we shall assume that only the desired hyperons get a VEV (and the rest of scalars may remain very heavy).Moreover, the SU (5) 3 framework also poses constraints on the possible family hypercharges of the hyperons, as collected in Appendix B. For the SU (5) 3 setup, it is convenient to add ϕ q13 ∼ (1, 1) (−1/6,0,1/6) , ϕ ℓ13 ∼ (1, 1) (1/2,0,−1/2) , (3.10) which are anyway required by the cyclic symmetry, to the set of hyperons which get a VEV.
The hyperons allow to write a set of non-renormalisable operators that provide effective Yukawa couplings for light fermions, as described in [1] by working in an effective field theory (EFT) framework.However, in our unified model, we need to introduce heavy messengers that mediate such effective operators in order to obtain a UV complete setup.For this, we add one set of vector-like fermions transforming in the 10 representation for each SU (5) factor, i.e. χ i ∼ 10 i and χ i ∼ 10 i .We shall assume that only the quark doublets Q i ∼ (3, 2) 1/6 i and Q i ∼ (3, 2) −1/6 i are relatively light and play a role in the effective Yukawa couplings, while the remaining degrees of freedom in χ i and χ i remain very heavy, We shall see that Q i and Q i also contribute to the RGEs in the desired way to achieve gauge unification.The full field content of this model also includes extra vector-like fermions Σ and Ξ as shown in Table 2.These play a role in the origin of neutrino masses as discussed in Section 3.2.
Finally, beyond the minimal set of Higgs doublets introduced in Section 2, we shall introduce here three pairs of 5, 5 and 45 Higgs representations preserving the cyclic symmetry.The doublets in the 5 and 45 mix, leaving light linear combinations that couple differently to down-quarks and charged leptons in the usual way [31], which we denote as H d i .Therefore, below the GUT scale we effectively have three pairs of Higgs doublets and H u,d 3 , such that the u-and d-labeled Higgs only couple to up-quarks (and neutrinos) and to down-quarks and charged leptons, respectively, in the spirit of the type II two Higgs doublet model.This choice is motivated to explain the mass hierarchies between the different charged sectors, as originally identified in [1], and could be enforced e.g. by a Z 2 discrete symmetry.We assume that the third family Higgs H u,d 3 are the lightest, they perform electroweak symmetry breaking and provide Yukawa couplings for the third family with O(1) coefficients if tan β ≈ 20.In contrast, we assume that the Higgs H u,d 1 , H u,d 2 have masses above the TeV (but much below the GUT scale) and act as messengers of the effective Yukawa couplings for the light families.
In detail, we assume that the SU (5) 3 group is broken down to the SM through the following symmetry breaking chain The SU (5) 3 breaking happens at the GUT scale, while the tri-hypercharge breaking may happen as low as the TeV scale, as allowed by current data [1], while the SM 3 breaking step is optional but may be convenient to achieve unification, and may be regarded as free parameter.This second breaking step is performed by the SU (3) i octets and SU (2) i triplets contained in Ω ij ∼ 24 i .See also Fig. 1 for an illustrative diagram.
We shall show that within this setup, achieving gauge unification just requires further assuming that three colour octets that live in Ω ij are light, while the remaining degrees of freedom of the bi-adjoints remain very heavy.Before that, we shall study in detail how our model explains the origin of the flavour structure of the SM.

Charged fermion mass hierarchies and quark mixing
The Higgs doublets in the cyclic 5 and 45 split the couplings of down-quarks and charged leptons in the usual way [31].We denote as H d i the linear combinations that remain light, with their effective couplings to down-quarks and charged leptons given by where (3.17) We focus now on the following set of couplings involving the hyperons, the vector-like fermions χ i and the light linear combinations of Higgs doublets, where i, j = 1, 2, 3, f u,d ij have mass dimension and the rest of the couplings are dimensionless.
After integrating out the heavy vector-like fermions χ i , χ i and Higgs doublets H u,d 1,2 , we obtain the following set of effective Yukawa couplings, where the dimensionless coefficients c u,d,e ij are given by It is clear that third family charged fermions get their masses from O(1) Yukawa couplings to the Higgs doublets H u,d 3 , where the mass hierarchies m b,τ /m t are explained via tan β ≈ λ −2 , where λ ≃ 0.224 is the Wolfenstein parameter.In contrast, quark mixing and the masses of first and second family charged fermions arise from effective Yukawa couplings involving the heavy messengers of the model, once the hyperons develop their VEVs.The heavy Higgs doublets H u,d 1 and H u,d 2 play a role in the origin of the family mass hierarchies, while the origin of quark mixing involves both the heavy Higgs and the vector-like quarks Q i and Q i , as shown in Fig. 2. We fix the various ⟨ϕ⟩ /M ratios in terms of the Wolfenstein parameter λ ≃ 0.224 We notice that the tiny masses of the first family are explained via the hierarchies of Higgs messengers in the spirit of messenger dominance [32].In other words, the heavy Higgs doublets H u,d 1 and H u,d 2 can be thought of gaining small effective VEVs from mixing with H u,d 3 , which are light and perform electroweak symmetry breaking, and these effective VEVs provide naturally small masses for light charged fermions.This is in contrast with the original spirit of tri-hypercharge, where the m 1 /m 2 mass hierarchies find their natural origin due to the higher dimension of the effective Yukawa couplings involving the first family [1].However, we note that in the SU (5) 3 framework, the three pairs of Higgs doublets H u,d i are required by the Z 3 symmetry, hence it seems natural that they play a role on the origin of fermion masses.Moreover, the introduction of these Higgs provides a very minimal framework to UV-complete the effective Yukawa couplings of tri-hypercharge, which otherwise would require a much larger amount of heavy messengers that are not desired, as they may enhance too much the RGE of the gauge couplings, eventually leading to a non-perturbative gauge coupling at the GUT scale.
The numerical values for the ratios in Eq. (3.24) provide the following Yukawa textures (ignoring dimensionless coefficients) where v SM is the usual SM electroweak VEV and the fit of the up-quark mass may be improved by assuming a mild difference between In general, the alignment of the CKM matrix is not predicted but depends on the choice of dimensionless coefficients and on the difference between Any charged lepton mixing is suppressed by the very heavy masses of the required messengers contained in χ i and χ i , leading to the off-diagonal zeros in Eq. (3.28), in such a way that the PMNS matrix must dominantly arise from the neutrino sector, as we shall see.We notice that a mild hierarchy of dimensionless couplings y e 1 /y d 1 ≈ λ 1.4 may be needed to account for the mass hierarchy between the down-quark and the electron.
The larger suppression of the (2,1), (3,1) and (3,2) entries in the quark Yukawa textures ensures a significant suppression of right-handed quark mixing.This is a very desirable feature, given the strong phenomenological constraints on right-handed flavour-changing currents [33,34].This way, we expect the model to reproduce the low energy phenomenology of Model 2 in [1], where the VEVs of the 23 and 13 hyperons may be as low as the TeV scale, while the VEVs of the 12 hyperons may be as low as 50 TeV or so.In this manner, we provide the following benchmark values for the mass scales involved in the flavour sector5

Neutrino masses and mixing
Explaining the observed pattern of neutrino mixing and mass splittings in gauge non-universal theories of flavour is usually difficult, due to the accidental U (2) 5 flavour symmetry predicted by these models, which is naively present in the neutrino sector as well.However, exotic variations of the type I seesaw mechanism have been shown to be successful in accommodating neutrino observations within non-universal theories of flavour, see Refs.[1,20].Here we will incorporate the mechanism of [1], which consists of adding SM singlet neutrinos which carry family hypercharges (although their sum must of course vanish).These neutrinos can be seen as the fermionic counterpart of hyperons, as they will connect the different hypercharge sites, therefore breaking the U (2) 5 flavour symmetry in the neutrino sector.In this manner, these neutrinos allow to write effective operators which may provide a successful pattern for neutrino mixing.However, the particular model presented in [1] incorporates SM singlet neutrinos with 1/4 family hypercharge factors, which cannot be obtained from SU (5) 3 , at least not from representations with dimension smaller than 45 6 according to a search with GroupMath [35].
Following the recipe of Ref. [1], we start by introducing two right-handed neutrinos: N atm ∼ (1, 1) (0,2/3,−2/3) and N sol ∼ (1, 1) (2/3,0,−2/3) , which will be responsible for atmospheric and solar neutrino mixing, respectively.These neutrinos are embedded in Σ 23 ∼ (1, 10, 10) and Σ 13 ∼ (10, 1, 10) representations of SU (5) 3 , respectively.We also need to add the cyclic permutation N cyclic embedded in Σ 12 ∼ (10, 10, 1) to preserve the cyclic symmetry of SU (5) 3 .However, we find that if the "cyclic" neutrino contained in Σ 12 is much heavier than the other neutrinos, then we can ignore it as it decouples from the seesaw, and we recover the minimal framework of Ref. [1].Finally, in order to cancel gauge anomalies, we choose to make these neutrinos vector-like by introducing the three corresponding conjugate neutrinos.The next step is adding hyperons that provide effective Yukawa couplings and Majorana masses for the singlet neutrinos.These are summarised in the Dirac and Majorana mass matrices that follow (ignoring the O(1) dimensionless couplings and the much heavier cyclic neutrinos) where the heavy scale M ξ is associated to the mass of the heavy vector-like fermions ξ 0 ∼ (1, 1) (0,0,0) , where we have defined ν as a 3-component vector containing the weak eigenstates of active neutrinos, while N and N are 2-component vectors containing the SM singlets N and conjugate neutrinos N , respectively.Now we assume that all the hyperons in Eqs.Dirac-type masses in m D L,R may be orders of magnitude smaller than the electroweak scale, because they arise from non-renormalisable operators proportional to the SM VEV.In contrast, the eigenvalues of M N are not smaller than O(v 23 ), which is at least TeV.Therefore, the condition m D ≪ M N is fulfilled in Eq. (3.37) and we can safely apply the seesaw formula to obtain, up to O(1) factors, This is the same texture that was obtained in Ref. [1], which is able to accommodate all the observed neutrino mixing angles and mass splittings [36,37] with O(1) parameters once the dimensionless coefficients implicit in Eq. (3.38) are considered.Remarkably, the singlet neutrinos N atm and N sol get masses around the TeV scale (v 23 ) and contribute to the RGE, while the cyclic neutrino is assumed to get a very heavy vector-like mass and decouples, as mentioned before.

Regime
Gauge group b i coefficients Table 3: b i coefficients of our model.See Appendix A for details on the gauge symmetries and particle content at each energy regime.

Gauge coupling unification
In order to ensure that the gauge couplings of our model do indeed unify into a single value at some high energy scale, we must solve their one-loop RGEs, which take the generic form [38] The b i coefficients depend on the specific group G i , with gauge coupling g i , and the representations in the model.They are given by Here µ is the renormalization scale, C 2 (G i ) is the quadratic Casimir of the adjoint representation of G i and S 2 (F i ) and S 2 (S i ) are the sums of the Dynkin indices of all fermion and scalar non-trivial representations under G i .Finally, κ = 1 (1/2) for Dirac (Weyl) fermions and η = 2 (1) for complex (real) scalars.
We computed the b i coefficients of our model, taking into account not only the gauge group for each energy regime, but also the particle content, since a particle decouples and does not contribute to the running at energies below its mass.The gauge symmetries and particle content at each energy regime are described in detail in Appendix A, whereas our results for the b i coefficients of the model are given in Table 3.Finally, we display results for the running of the gauge couplings in Before discussing gauge coupling unification, we note that we expect radiative corrections to disturb the scales above unless some couplings in the scalar potential are fine-tuned to some extent.In z Navarro Flavour deconstruction from the EW scale to the GUT scale 13 Gauge coupling unification ies due to gauge coupling nditions: < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 V q H U 5 K c p V T c d i i B P n r B B 6 a K e X k = " > A I r e j C f j x X g 3 P q a l S 8 a s 5 w D 9 g f H 1 A 3 s 0 o h 0 = < / l a t e x i t >

U (1) Y3
< l a t e x i t s h a 1 _ b a s e 6 4 = " U L J r c g R k S / e u 3 I T e j e e Y l G S A o 3 I 3 L y X c g t i K 4 v P 6 j J B C f C x J p g I p u + 3 y A A L T E C H n A X j r M a w T p q V s l M t V x 8 q x d r t I q I 8 O k P n q I Q c d I 1 q 6 B 7 V U Q M R 9 I L e 0 D v 6 M F 6 N q f F p f M 1 b c 8 Z i 5 h Q t w f j 5 B R Z i q o k = < / l a t e x i t > U (1) Y12 < l a t e x i t s h a 1 _ b a s e 6 4 = " k p D g k q P r X U i s 0 5 h x j 4 l G q 5 l Y 5 T 4 A A e c g y a 4 A S 3 Q B h g 8 g m f w C t 6 M J + P F e D c + Z q 1 L R j l z A P 7 I + P w B 5 f y d b Q = = < / l a t e x i t > help bend SU(2).
< l a t e x i t s h a 1 _ b a s e 6 4 = " t Q J 4 / 3 T b / t e i F 4 n 8 x N b 7 n + V h W P x 6 D + V N w G p 3 u 9 w U H v 4 P N e + / D t 6 h n Z 8 l 5 4 L 7 1 X 3 s B 7 7 R 1 6 R 9 6 x N / K 4 9 9 3 7 4 f 3 0 f j V + N / 4 2 m 8 3 N a + q 9 j d W c 5 1 7 F m q 1 / H 6 v C l w = = < / l a t e x i t > i < l a t e x i t s h a 1 _ b a s e 6 4 = " e y p S U l N k P o 0  particular, the bi-adjoint scalars Ω ij may couple at tree-level to the light Higgs doublets in order to give corrections to their masses proportional to the GUT scale.This is a consequence of the well-known hierarchy problem that afflicts all non-supersymmetric GUTs.To avoid this, perhaps one could build a SU (5) 3 tri-unification supersymmetric theory, but this would lead to rapid proton decay via d = 5 operators mediated by the coloured Higgs triplet superpartners 7 .A solution to this would require further model building [6,9].Another option would be to include a strongly coupled sector to generate the light scalars as pseudo Goldstone bosons, e.g. one could introduce a strongly coupled SU (5) [5] and generalise the permutation symmetry to Z 4 in order to enforce a single gauge coupling.This would require further model building beyond the scope of this paper.Nevertheless, without the need of imposing supersymmetry nor extra dynamics, the nine gauge couplings of the SM 3 group unify at a very high unification scale M GUT ≈ 10 17 GeV, slightly above the SM 3 breaking scale, with a unified gauge coupling g GUT ≈ 1.44.We note the important role played by three Θ i colour octets embedded into Ω ij , and by the Q i vector-like quarks which also act as heavy messengers of the flavour theory, which are crucial to modify the running of the SU (3) and SU (2) gauge couplings in order to achieve unification.We also remark that the discontinuities in Figs. 4 and 5 are due to the gauge coupling matching conditions that apply at the steps in which the U (1) Y group is decomposed into two (first discontinuity) and three hypercharges (second Even though the Z 3 symmetry gets broken at the SM 3 breaking scale, it stays approximate at low energies, down to the tri-hypercharge breaking scale, and only the running of U (1) Y 3 is slightly different from that of the other two hypercharge groups.In fact, the gauge couplings of the U (1) Y 1 and U (1) Y 2 groups almost overlap and cannot be distinguished in Fig. 4.This can be easily understood by inspecting the b i coefficients on Table 3.Then, the matching conditions at v 12 = 50 TeV split the low energy g Y 12 and g Y 3 couplings, which become clearly different: g Y 12 (v 12 ) ≈ 0.59 and g Y 3 (v 12 ) ≈ 0.79.Finally, at v 23 = 5 TeV one recovers the standard SU (3) c × SU (2) L × U (1) Y gauge group, which remains unbroken down to the electroweak scale.
In order to study how unification changes with the scale of SM 3 breaking, v SM 3 , we consider the values v SM 3 = 5 • 10 15 GeV and v SM 3 = 5 • 10 17 GeV and fix the rest of the intermediate scales as in Eq. (3.41).Results for the running of the gauge couplings in these two scenarios are shown in Fig. 5. On the left-hand side we show the case v SM 3 = 5 • 10 15 GeV whereas on the right-hand side we display our results for v SM 3 = 5 • 10 17 GeV.In the first case, our choice of SM 3 breaking scale leads to unification of the gauge couplings at a relatively low scale, M GUT ≈ 1.8 • 10 16 GeV.This is potentially troublesome, as it may lead to too fast proton decay, as explained below.In contrast, when the SM 3 breaking scale is chosen to be very high, as in the second scenario, unification also gets delayed to much higher energies.In fact, we note that with our choice v SM 3 = 5 • 10 17 GeV, gauge coupling unification already takes place at the SM 3 breaking scale, M GUT ≈ v SM 3 .In this case, SU (5) 3 breaks directly to the tri-hypercharge group and there is no intermediate SM 3 scale.Finally, the impact of v SM 3 is further illustrated in Fig. 6.
Here we show the relation between M GUT , v SM 3 and α −1 GUT = 4π/g 2 GUT .These two plots have been made by varying v SM 3 and all the other intermediate scales fixed as in Eq. (3.41).The left-hand side of this figure confirms that larger v SM 3 values lead to higher unification scales and smaller gaps between these two energy scales.The right-hand side of the figure shows the relation between the unified gauge coupling and the GUT scale.Again, the larger M GUT (or, equivalently, larger v SM 3 ) is, the larger g GUT (and smaller α −1 GUT ) becomes.In particular, in this plot g GUT ranges from ∼ 1.30 to ∼ 1.53.

Proton decay
As in any GUT, proton decay is a major prediction in our setup.In standard SU (5) the most relevant proton decay mode is usually p → e + π 0 .This process is induced by the tree-level exchange of the X gauge bosons contained in the 24 (adjoint) representation, such as the (3, 2) 5 6 vector leptoquark.
Integrating out these heavy vector leptoquarks leads to effective dimension-6 operators8 that violate both baryon and lepton numbers, for instance qqqℓ.The resulting proton life time can be roughly estimated as where m X is the mass of the heavy leptoquark, m p ≈ 0.938 GeV is the proton mass and α GUT = g 2 GUT /(4π) is the value of the fine structure constant at the unification scale.For a comprehensive review on proton decay we refer to [41].
However, given that in our setup the three generation leptoquarks get the same mass, in practice the gauge leptoquark phenomenology is that of conventional (flavour universal) SU (5).One can easily estimate that for m X = 10 17 GeV and g GUT ∼ 1.5, the proton life time is τ p ∼ 10 38 years, well above the current experimental limit, τ (p → e + π 0 ) > 2.4 • 10 34 years at 90% C.L. [39].Therefore, a large unification scale suffices to guarantee that our model respects the current limits on the proton lifetime.In fact, such a long life time is beyond the reach of near future experiments, which will increase the current limit by about one order of magnitude [40].
A more precise determination of the p → e + π 0 decay width is [41,42] where A L ≈ 1.247 accounts for the QCD RGE from the M Z scale to m p [41].In contrast, A SL(R) accounts for the short-distance RGE from the GUT scale to M Z , given by where b i and γ i denote the β coefficients and the anomalous dimensions computed at one-loop in Tables 3 and 4 where the errors (shown in the parenthesis) denote statistical and systematic uncertainties, respectively.Given that in our model we have three SU (5) groups, we actually have three generations of the usual SU (5) leptoquarks, coupling only to their corresponding family of chiral fermions.However, since the three SU (5) i groups are all broken down to their SM i subgroups at the same scale, in practice the model reproduces the phenomenology of a flavour universal leptoquark as in conventional SU (5), albeit with the specific fermion mixing predicted by our model as shown in where Notice that even though our flavour model predicts non-generic fermion mixing, the alignment of the CKM matrix is not univocally predicted but relies on the choice of dimensionless coefficients.Assuming the CKM mixing to originate mostly from the down sector we find C L ≃ 1.946 and C R ≃ 0.999, while if the CKM mixing originates mostly from the up sector we find very similar coefficients as C L ≃ 1.946 and C R ≃ 0.974.Therefore, the prediction for proton decay is robust and independent of the alignment of the CKM to excellent accuracy.We show our numerical results for the p → e + π 0 lifetime in Fig. 7. Again, v SM 3 varies in the left panel of this figure, while the rest of intermediate scales have been chosen as in Eq. (3.41).The right panel shows an equivalent plot with the p → e + π 0 lifetime as a function of M GUT .This figure provides complementary information to that already shown in Fig. 6.In both cases we have used the precise determination of the lifetime in Eq. (3.43), but we note that the estimate in Eq. (3.42) actually provides a very good approximation, with τ p /τ app p ∈ [0.5, 1.2] in the parameter region covered in Fig. 7.The current Super-Kamiokande 90% C.L. limit on the p → e + π 0 lifetime, τ (p → e + π 0 ) > 2.4 • 10 34 years [39], excludes values of the GUT scale below M GUT ∼ 1.3 • 10 16 GeV, while the projected Hyper-Kamiokande sensitivity at 90% C.L. after 20 years of runtime, τ (p → e + π 0 ) > 1.2 • 10 35 years [40], would push this limit on the unification scale in our model to M GUT ∼ 2 • 10 16 GeV.Therefore, our model will be probed in the next round of proton decay searches, although large regions of the parameter space predict a long proton lifetime, well beyond any foreseen experiment.

Conclusions
In this paper we have discussed SU (5) 3 with cyclic symmetry as a possible GUT.The basic idea of such a tri-unification is that there is a separate SU (5) for each fermion family, with the light Higgs doublet(s) arising from the third family SU (5), providing a basis for charged fermion mass hierarchies.We have set out a general framework in which a class of such models which have been proposed in the literature, including U (1) 3 Y , SU (2) 3 L and other related models, may have an ultraviolet completion in terms of SU (5) 3 tri-unification.
The main analysis in the paper was concerned with a particular embedding of the tri-hypercharge model U (1) 3 Y into SU (5) 3 with cyclic symmetry.We showed that a rather minimal example can account for all the quark and lepton (including neutrino) masses and mixing parameters.This same example can also satisfy the constraints of gauge coupling unification into the cyclic SU (5) 3 gauge group, by assuming minimal multiplet splitting, together with a set of relatively light colour octet scalars.The approximate conservation of the cyclic symmetry at low energies is also crucial to achieve gauge unification.The heavy messengers required to generate the flavour structure also modify the RGE in the desired way, highlighting the minimality of the framework.
Finally, we have also studied proton decay in this example, and presented the predictions of the proton lifetime in the dominant e + π 0 channel.The results depend on the scale at which the three SM gauge groups break down into their diagonal non-Abelian subgroup together with tri-hypercharge, which is a free parameter in this model, enabling the proton lifetime to escape the existing Super-Kamiokande bound, but be possibly observable at Hyper-Kamiokande.In this manner, the signals on proton decay may allow to test the model at high scales, while low energy signals associated with tri-hypercharge enable the model to be tested by collider and flavour experiments.We conclude that SU (5) 3 tri-unification reconciles the idea of gauge non-universality with the idea of gauge coupling unification, opening up the possibility to build consistent non-universal descriptions of Nature that are valid all the way up to the scale of grand unification.

A Energy regimes, symmetries and particle content
We describe the symmetries and particle content of our model at each energy regime between the GUT and electroweak scales.As a result of SU (5) 3 breaking, each of the fermion representations F i and T i becomes charged under an SU (3) × SU (2) × U (1) factor.Regarding the rest of the fields, most get masses at the M GUT ∼ v GUT unification scale and decouple.We will assume that only those explicitly required at low energies remain light.For instance, out of all the components of the Ω ij scalars, only the Θ i and ∆ i states, belonging to the adjoint representations of SU (3) i and SU (2) i , respectively, remain in the particle spectrum.Similarly, only some SM singlets in the Φ i scalar fields are assumed to be present at this energy scale.For instance, this is the case of Φ ℓ23 , contained in Φ  8.These representations eventually become the tri-hypercharge hyperons at lower energies.Similarly, the Q i vector-like quarks in the χ i and χ i multiplets are also assumed to be present at this energy scale.The full fermion and scalar particle content of the model in this energy regime is shown in Table 5.The gauge couplings above (g s i and g L i , with i = 1, 2, 3) and below (g s and g L ) the breaking scale verify the matching relations g s 1 g s 2 g s 3 g 2 s 1 g 2 s 2 + g 2 s 1 g 2 s 3 + g 2 s 2 g 2 s 3 = g s , (A.50) = g L , (A.51) which are equivalent to with α −1 i = 4π/g 2 i .The main difference with respect to the original tri-hypercharge model [1] is that a complete ultraviolet completion for the generation of the flavour structure is provided in our setup.As already explained, we achieve this with the hyperons and vector-like fermions present in the particle spectrum, which originate from SU (5) 3 representations.We assume N cyclic as well as the conjugate representation N cyclic to be decoupled at this energy scale.Similarly, the ∆ i triplets are also assumed to get masses of the order of the SM 3 breaking scale and decouple.The resulting fermion and scalar particle content of the model is shown in Table 6.

Regime 3: ξ scale → H 1 scale
The next energy threshold is given by the ξ singlets, responsible for the flavour structure of the neutrino sector, with masses M ξ ∼ 10 10 GeV.At this scale, the ξ 0 as well as the ξ 12 , ξ 13 , ξ 23 and their conjugate representations are integrated out and no longer contribute to the running of the gauge couplings.The gauge symmetry does not change and stays the same as in the previous energy regime.The resulting particle spectrum is that of Table 6 removing the ξ singlet fermions., the Q i vector-like quarks and the Θ i colour octets decouple from the particle spectrum of the model.As in the previous two energy thresholds, the gauge symmetry is not altered.The particle spectrum at this stage is that shown on Table 6 removing the ξ singlet fermions, the H u,d 1,2 scalar doublets, the Q i vector-like quarks and the Θ i colour octets.Hyperons are responsible for the breaking of the tri-hypercharge symmetry.In a first hypercharge breaking step, U which is equivalent to (A.56) The "12 hyperons" ϕ ℓ12 , ϕ q12 and ϕ u12 get masses of the order of ⟨ϕ q12 ⟩ and decouple at this stage.We also assume the Θ i colour octets to be integrated out at the tri-hypercharge breaking scale.The resulting fermion and scalar particle content is shown in Table 7.

Figure 1 :
Figure 1: Diagram showing the different scales of spontaneous symmetry breaking in our example model (see also Eqs.(3.12-3.15)),along with the accidental, approximate flavour symmetries (U (3) 5 and U (2) 5 ) that arise at low energies.

Figure 2 :
Figure 2: Diagrams in the model which lead to the origin of light charged fermion masses and quark mixing, where i = 1, 2.

Figure 3 :
Figure 3: Diagrams leading to effective Yukawa couplings in the neutrino sector.
plus cyclic permutations), which are embedded in the representations Ξ 0 ∼ (1, 1, 1) and Ξ 23 ∼ (1, 5, 5) (plus conjugate, plus cyclic permutations) of SU (5) 3 .Example diagrams are shown in Fig 3. We now construct the full neutrino mass matrix as (3.35-3.36)get VEVs at the scale v 23 of 23 hypercharge breaking according to Eq. (3.14), and we have into account that ⟨ϕ q13 ⟩ / ⟨ϕ q23 ⟩ ≈ λ as obtained from the discussion of the charged fermion sector in Section 3.1.It is also required to assume M Natm , M Natm v 23 in order to obtain the observed neutrino mixing with the textures of Eqs.(3.35-3.36).

Fig. 4 . 2 = 1 = 4 •
This figure has been obtained by fixing the intermediate energy scales to v 23 = 5 TeV , M Θ = 100 TeV , M ξ = 10 10 GeV , v 12 = 50 TeV , M H u,d 400 TeV, v SM 3 = 6 • 10 16 GeV .M Q = 100 TeV , M H u,d 10 4 TeV , (3.41) j y W s z 4 z f Z 0 + x 2 e 4 E + z X a c Z h C 7 o Q d s v n P 0 S T o 6 5 5 P C V A r r e r 2 b j c 0 H j e b W w + 0 d / 9 H j J 0 + f 7 e 4 9 P 7 M 6

r b 1 XFigure 4 : 2 = 1 = 4 • 10 4
Figure 4: Running of the gauge couplings.The red lines correspond to the SU (3) gauge couplings, the blue ones to the SU (2) gauge couplings and the black/grey ones to the U (1) gauge couplings.A zoom-in with the high-energy region close to the unification scale is also shown.These results have been obtained with v 23 = 5 TeV, v 12 = 50 TeV, M Q = 100 TeV, M H u,d 2 = 400 TeV, M H u,d 1 = 4 • 10 4 TeV, M ξ = 10 10 GeV and v SM 3 = 6 • 10 16 GeV.The discontinuities in the plot are due to the gauge coupling matching conditions that apply at each symmetry breaking step, see main text and Appendix A.

Figure 5 :
Figure 5: Running of the gauge couplings.Colour code as in Fig. 4.These results have been obtained with v SM 3 = 5 • 10 15 GeV (left) and v SM 3 = 5 • 10 17 GeV (right).The rest of the intermediate scales have been chosen as in Eq. (3.41).The discontinuities in the plots are due to the gauge coupling matching conditions that apply at each symmetry breaking step, see main text and Appendix A

Figure 6 :
Figure 6: M GUT as a function of v SM 3 (left) and α −1 GUT (right).The SM 3 breaking scale v SM 3 varies in these plots, while the rest of the intermediate scales have been fixed to the values in Eq. (3.41).The shaded grey region is excluded by the existing Super-Kamiokande 90% C.L. limit on the p → e + π 0 lifetime, τ (p → e + π 0 ) > 2.4 • 10 34 years [39], whereas the horizontal dashed line corresponds to the projected Hyper-Kamiokande sensitivity at 90% C.L. after 20 years of runtime, τ (p → e + π 0 ) > 1.2 • 10 35 years, obtained in [40].See Section 3.4 for details on the proton decay calculation.Finally, the shaded yellow region on the left-hand plot is excluded due to M GUT ≤ v SM 3 .

Figure 7 :
Figure 7: τ (p → e + π 0 ) as a function of v SM 3 (left) and M GUT (right).The SM 3 breaking scale v SM 3 varies in these plots, while the rest of the intermediate scales have been fixed to the values in Eq. (3.41).The choice v SM 3 = 5 • 10 17 GeV, which leads to M GUT ≈ v SM 3 , is highlighted with a red point in both plots.The shaded grey region is excluded by the existing Super-Kamiokande 90% C.L. limit on the p → e + π 0 lifetime, τ (p → e + π 0 ) > 2.4 • 10 34 years [39], whereas the horizontal dashed line corresponds to the projected Hyper-Kamiokande sensitivity at 90% C.L. after 20 years of runtime, τ (p → e + π 0 ) > 1.2•10 35 years, obtained in [40].

Regime 4 : 1 ∼ 10 4
H 1 scale → H 2 scaleAt energies of the order of M H u,d TeV, the H u,d 1 scalar doublets decouple from the particle spectrum of the model.Again, the gauge symmetry does not change.The particle spectrum at this stage is that shown on Table6removing the ξ singlet fermions and the H u,d 1 scalar doublets.

Regime 5 : 2 ∼Regime 6 : 1 ) Y 1 × U ( 1 ) Y 2 × U ( 1 ) Y 3
H 2 scale → Q, Θ scale At energies of the order of M H u,d 100 TeV, the H u,d 2 scalar doublets decouple from the particle spectrum of the model.As in the previous two energy thresholds, the gauge symmetry remains the same.The particle spectrum at this stage is that shown on Table 6 removing the ξ singlet fermions and the H u,d 1,2 scalar doublets.Q, Θ scale → SU (3) c × SU (2) L × U (breaking scale At M Q ≲ M H u,d 2

Table 1 :
Minimal content for the general SU (5) 3 setup.Due to the cyclic symmetry, there are only four independent representations, one for each of the fermions F i , T i and the scalars Ω ij , H i .

Table 2 :
Fermion and scalar particle content and representations under SU (5) 3 .F i and T i include the chiral fermions of the SM in the usual way, while χ i , ξ's and Ξ's (highlighted in yellow) are vector-like fermions, thus the conjugate partners must be considered.Ω's, H's and Φ's are scalars.

Table 6 :
Fermion and scalar representations under SU (3) Y3 in energy regimes 2, 3, 4, 5 and 6.Some states in this table get decoupled at intermediate scales and are not present at all energy regimes, see text for details.Fermions highlighted in yellow belong to a vector-like pair and thus have a conjugate representation not shown in this table.

Table 7 :
Fermion and scalar representations underSU (3) c × SU (2) L × U (1) Y12 × U (1) Y3 in energy regime 7.Fermions highlighted in yellow belong to a vector-like pair and thus have a conjugate representation not shown in this table.