Minimal complete tri-hypercharge theories of flavour

The tri-hypercharge proposal introduces a separate gauged weak hypercharge assigned to each fermion family as the origin of flavour. This is arguably one of the simplest setups for building"gauge non-universal theories of flavour"or"flavour deconstructed theories". In this paper we propose and study two minimal but ultraviolet complete and renormalisable tri-hypercharge models. We show that both models, which differ only by the heavy messengers that complete the effective theory, are able to explain the observed patterns of fermion masses and mixings (including neutrinos) with all fundamental coefficients being of $\mathcal{O}(1)$. In fact, both models translate the complicated flavour structure of the Standard Model into three simple physical scales above electroweak symmetry breaking, completely correlated with each other, that carry meaningful phenomenology. In particular, the heavy messenger sector determines the origin and size of fermion mixing, which controls the size and nature of the flavour-violating currents mediated by the two heavy $Z'$ gauge bosons of the theory. The phenomenological implications of the two minimal models are compared. In both models the lightest $Z'$ remains discoverable in dilepton searches at the LHC Run 3.


Introduction
The Standard Model (SM) of particle physics, though highly successful, offers no insight into the origin of the three fermion families nor the pattern of fermion Yukawa couplings, which results in a hierarchical mass spectrum for quarks and charged leptons that may be expressed at low energies as [1] ) ) where v SM ≃ 246 GeV is the SM vacuum expectation value (VEV) and λ ≃ 0.224 is the Wolfenstein parameter which parameterises the CKM matrix as (1.4) In addition, the discovery of very small neutrino masses and large PMNS mixing [2,3], has made the flavour puzzle difficult to ignore.A tentative approach to explain the flavour puzzle consists of embedding the SM in a larger gauge symmetry that contains a separate gauge group for each fermion family, with the light Higgs doublet(s) originating from the third family group.This general idea is sometimes denoted in the literature as either "gauge non-universal theories of flavour" or "flavour deconstructed theories".Different possibilities have been discussed since the early 80s [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18], with new ideas appearing also in recent times [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36].Remarkably, these theories distinguish the three fermion families (hence avoiding family replication) and are able to explain the flavour structure of the SM with all fundamental couplings being of O (1).Out of them, arguably the simplest one involves just assigning one separate gauged weak hypercharge to each fermion family [30] SU (3) (1.6) Under the extended gauge group in Eq. (1.6) the i-th fermion family only carries Y i hypercharge, with the other hypercharges set equal to zero (see Table 1), where Y = Y 1 + Y 2 + Y 3 is equal to the usual SM weak gauged hypercharge.Anomalies cancel separately for each family, as in the SM, but without family replication.Two Higgs doublets are introduced which carry third family hypercharge, H u,d 3 ∼ (1, 2) (0,0,±1/2) , leading to renormalisable Yukawa couplings only involving the third family.Yukawa couplings involving the first and second families are generated only after the gauge symmetry is broken to the SM gauge group, leading to a hierarchical pattern of Yukawa couplings.
The so-called "tri-hypercharge" (TH) gauge group in Eq. (1.6) is broken down to the SM gauge group by the VEVs of appropriate scalars that carry family hypercharges which add up to zero.

The breaking chain
(1.9) leads to a successful generation of charged fermion mass hierarchies and quark mixing.There generally exists a hierarchy between v 12 and v 23 that is related to the origin of the fermion mass hierarchies, as we shall see.Typically In [30] two of us provided a general recipe for model building in the tri-hypercharge framework based on a spurion formalism, we proposed simple models in an effective field theory (EFT) framework (without specifying the heavy messengers that complete the theory) and studied the main phenomenological predictions of the setup, which are connected to heavy gauge bosons that arise after the spontaneous breaking of the TH group.Then, in [32] the three of us built an ultraviolet (UV) complete tri-hypercharge theory that arises from a gauge unified framework based on assigning a separate SU (5) gauge group to each fermion family, with the three SU (5)s related by a cyclic Z 3 permutation symmetry that enforces a single gauge coupling at the grand unification (GUT) scale.
In this paper we provide two examples of minimal but ultraviolet complete tri-hypercharge models that contain the minimal amount of total degrees of freedom and representations, being amongst the most minimal models in the literature .In particular, the rank of the trihypercharge symmetry is the most minimal.We show that both models, which differ only by the heavy messengers that complete the effective theory, are able to explain the observed patterns of fermion masses and mixings (including neutrinos) with all fundamental coefficients being of O (1).
In fact, we shall see that the two minimal models presented here translate the complicated flavour structure of the Standard Model into three simple physical scales above electroweak symmetry breaking, completely correlated with each other and with the flavour hierarchies of the SM, that carry meaningful phenomenology.In this regard, we find that the heavy messengers crucially dictate the origin and size of fermion mixing, which controls the size and nature of the flavour-violating currents mediated by the two heavy Z ′ gauge bosons of the theory.The phenomenological implications of the two minimal models are compared, where the lightest Z ′ is discoverable in dilepton searches at the LHC Run 3 in both cases.
The layout of the paper is as follows.In Section 2 we first introduce the tri-hypercharge theory from a bottom-up perspective and an EFT point of view, and then we show two minimal UVcomplete models.In the model of Section 2.1 all the heavy messengers that complete the effective theory are vector-like fermions (hence having the simplest scalar sector and potential), whereas the model of Section 2.2 offers a combination of heavy Higgs doublets and vector-like fermions that contains the minimal amount of degrees of freedom and representations.In Section 3 we discuss the neutrino sector that is common for both complete models.Then in Section 4 we discuss the model-dependent phenomenology associated to the two complete models, which is mostly dominated by flavour-changing neutral currents (FCNCs).Finally, Section 5 concludes the paper.Additional technical details are given in three separate appendices.
Table 1: Field content of the low-energy (EFT) tri-hypercharge theory.q i and ℓ i (where i = 1, 2, 3) are left-handed (LH) SU (2) L doublets of chiral quarks and leptons, while u c i , d c i and e c i are the CP conjugate 1 righthanded (RH) quarks and leptons (so that they become left-handed).Two singlet neutrinos ν c 1,2 , also taken as left-handed fields, will be relevant for the origin of neutrino masses.H u,d 3 are SU (2) L Higgs doublets, while ϕ q12,ℓ12,q23,ℓ23 are the so-called scalar hyperons which spontaneously break the three gauge hypercharges down to SM hypercharge (the diagonal subgroup

Minimal Models
We consider that at relatively high energies the gauge symmetry contains one hypercharge abelian factor for each fermion family.These are spontaneously broken via the VEVs of ϕ scalars (see Table 1) carrying non-zero family hypercharges that add up to zero, so that they are gauge singlets under the SM symmetry, and are denoted as hyperons, 1 For a given fermion Ψ that consists of two 4-component chiral spinors ΨL = (ψL, 0) T and ΨR = (0, ψR) T , where ψL and ψR are 2-component (Weyl) spinors, one may get rid of chiral indices by defining ψ ≡ ψL and by taking the CP conjugate of ψR, i.e. ψ c ≡ CP ψR(CP ) −1 , which is by construction a left-handed field.Then one can write the theory in terms of ψ and ψ c only.For example, fermion bilinears among both notations are related by ΨRΨL → ψ c ψ.We also refer the reader to [37,38] for extensive reviews on 2-component spinor notation and techniques. (2.12) At renormalisable level, only third family Yukawa couplings are allowed.Two (third family) Higgs doublets H u,d 3 ∼ (1, 2) (0,0,±1/2) perform spontaneous breaking of the electroweak symmetry, and explain the mass hierarchies m b,τ /m t if we assume a type-II two Higgs doublet model (2HDM) with tan The masses of the light charged fermions arise from non-renormalisable operators involving the hyperon scalars.With the four2 hyperons of Table 1, one can generate all the spurions3 introduced in Ref. [30], which populate the Yukawa matrices.By working in an EFT framework, we can write (ignoring O(1) dimensionless coefficients), Notice that the 23-hyperons generate the mass hierarchy m 2 /m 3 along with naturally suppressed V cb and V ub CKM elements.Then, the 12-hyperons discriminate the first family, generating the mass hierarchy m 1 /m 2 , the Cabbibo angle V us and an extra suppression for V ub with respect to V cb , while the origin of small neutrino masses and PMNS mixing is later discussed in the complete model, see Section 3.
In the following we UV-complete the above EFT by including heavy messengers explicitly.Their masses provide the heavy scales Λ 1 and Λ 2 .Notice that we do not need to generate all the entries in the Yukawa matrices of Eqs.(2.13-2.15) to explain the flavour structure of the SM, and indeed some of them will not be generated at tree-level in our complete models.
In the next two Sections we will show two minimal UV completions, one where all the heavy messengers are vector-like (VL) fermions (hence having the simplest scalar sector and potential), and another one with a combination of heavy Higgs doublets and VL fermions that contains the minimal amount of degrees of freedom and representations.

Model 1
Our first proposal for a minimal, complete and renormalisable model includes three vector-like SU (2) L singlet fermions for each charged sector, as shown in Table 2, that act as heavy messengers of the effective Yukawa operators introduced before.The full set of renormalisable Yukawa couplings and bare mass terms in the charged fermion sector of the model is given by ) After integrating out the heavy VL charged fermions, the effective Yukawa couplings for chiral charged fermions take the form   1, are displayed, while their vector-like partners with conjugate quantum numbers are implicitly assumed to be present but not explicitly shown.small Yukawa couplings for light charged fermions.Now we shall fix the ratios of hyperon VEVs over messenger masses in order to explain the flavour structure of the SM.Firstly, notice how the 23-messengers always appear together with the 23-hyperons in the Yukawa matrices, and similarly the 12,13-messengers which distinguish the first family always appear together with 12-hyperons.This suggests the presence of one scale at which all the 23messengers live -that we associate with Λ 2 from the EFT formalism -and another, heavier scale where all the 12,13-messengers live -that we associate with Λ 1 .The exception is the ratio ϕ ℓ23 /M 12 appearing in the (1,1) entries, which provides a connection between both scales.This is a highly non-generic feature in this class of models, which was not anticipated by the EFT framework.
In fact, assuming for simplicity the two 23-VEVs to be degenerate and denoted generically as ⟨ϕ 23 ⟩, the two 12-VEVs to be degenerate and denoted as ⟨ϕ 12 ⟩, the 23-messenger masses to be degenerate and denoted as M 23 and the 12,13-messenger masses to be degenerate and denoted as M 12,13 , then we find that the approximate scaling provides an excellent explanation of the SM flavour structure, which is translated to three simple and physical scales4 of new physics (NP) above electroweak symmetry breaking, as shown in Fig. 2. Indeed, we can explain the whole flavour structure of the SM in terms of just three small parameters, The third ratio ⟨ϕ 23 ⟩/M 12,13 ∼ λ 5 , which only appears in the (1,1) entry of the Yukawa matrices, provides an extra (welcomed) suppression for first family fermion masses.Moreover, this predicts a highly non-generic 5 relation between the two VEV scales as follows (up to O(1) variation), and also, together with the first ratio of Eq. ( 2.24), it suggests that M 23 ∼ v 12 .A typical benchmark compatible with experimental searches then is, as depicted in Fig. 2, where, thanks to the non-generic relation of Eq. (2.25), testing experimentally any of these scales effectively tests all of them (and in particular, setting bounds on any of them also sets bounds on the rest).Notice that our model then translates the complicated flavour structure of the SM into just three simple and physical scales of new physics above electroweak symmetry breaking, as shown in Fig. 2.However, we do not expect the masses of different messengers nor the VEVs of different hyperons to be exactly degenerate -they may naturally vary by O(1) factors.Therefore, we take advantage of these natural O(1) variations with respect to Eq. (2.24) to explain the slightly more accentuated mass hierarchies of the up sector and the tinier mass of the electron with respect to first family quarks.More specifically, we assume the following numerical values for the ratios appearing in the Yukawa matrices, which are consistent with O(1) variations from the spectrum of Fig. 2 (or similarly O(1) variations from the three fixed ratios of Eq. (2.24)), ) ) and the (1,1) entries of the effective Yukawa matrices are further suppressed by fixed ratios that involve 23-VEVs in the numerator and 12-masses in the denominator, We then predict the approximate numerical textures (up to O(1) coefficients) for the effective Yukawa couplings Flavour deconstruction from the EW scale to the GUT scale Tri-hypercharge: spectrum < l a t e x i t s h a 1 _ b a s e 6 4 = " 0 T 4 V 8 q w a M w x d q < l a t e x i t s h a 1 _ b a s e 6 4 = " K F 5 l 4 u J 9 x X 1 N n e W z j j K x f t l K o 8 e V m 0 a 1 W r X q 3 f 1 S q N q y K O E j p G J + g M W e g C N d A t a q I W I u g R P a N 3 9 K E 9 a a / a p / Y 1 a 1 3 R i p k j N F f a 9 w + / W q j 1 < / l a t e x i t > Model 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " i 5 Model 2 < l a t e x i t s h a 1 _ b a s e 6 4 = " h A < l a t e x i t s h a 1 _ b a s e 6 4 = " E h 2 S j 0 5 G 7 x r e y s r q 2 v r F Z 2 V K 3 d 3 b 3 9 r W D w y 6 P U 4 a w j W I a s 7 4 H O a Y k w r Y g g u J + w j A M P Y p 7 3 u S q q P c e M O M k j u 7  + q 1 q 2 q 3    Notice that according to the above textures, CKM mixing in this model originates mostly from the down sector.However, if one assumes for simplicity degenerate VEVs and masses as in Eq. (2.24), then the Yukawa textures would be similar (up to an overall normalisation factor) in both the up and down sectors -then the alignment of the CKM depends on the off-diagonal c u,d ij coefficients, as in the model of Ref. [32].In this case, all the coefficients c u,d,e ij would still be of O(1), with the exception of c u 11 and c d 11 which would need to be of O(0.1) and of O(4), respectively.This is completely acceptable as c u,d 11 are given as a product of three fundamental couplings in the complete theory.
On the other hand, the prediction of significant 1-2 RH quark mixing, especially in the down sector, is of great importance for the phenomenology of the heaviest Z ′ 12 boson.Similarly, the presence of CKM-like mixing in the charged lepton sector implies that the two Z ′ bosons of the model potentially mediate charged lepton flavour-violating processes (CLFV) at acceptable rates, as discussed later in Section 4. Most of the PMNS mixing is however generated from the neutrino sector, as discussed later in Section 3.

Model 2
Now we envisage an alternative complete and renormalisable model which contains less VL fermion messengers by exchanging several of them by heavy Higgs doublets.Here we exchange the VL

fermions of D-type and E-type by the pair of Higgs doublets
, as shown in Table 3.These messengers are motivated by the tri-hypercharge model of Ref. [32] that originates from a gauge unified framework.However, we still need to consider VL quarks to generate the effective Yukawa operators that provide quark mixing.We could use the VL quark doublets as in [32], with i = 1, 2, 3. Nevertheless, here we shall stick with the more minimal set of up-type VL quark singlets U 12 , U 13 and U 23 that were already present in Model 1.On top of generating quark mixing, the U ij VL quarks will also generate the small masses of light up-quarks, and will provide unique phenomenological predictions as we shall see in Section 4.
With respect to Model 1, Model 2 contains a smaller number of total degrees of freedom and representations, but also more scalars fields and hence more terms in the scalar potential.We refer to Appendix A, where we discuss the scalar potential of the two models and briefly comment as well on the stability of the spectrum in Fig. 2 under radiative corrections.
We remark that the full set of renormalisable Yukawa couplings in the up sector is also described in this model by Eq. (2.16).In contrast, the renormalisable Yukawa couplings in the down and charged lepton sector are simply given by Notice however that the Higgs doublets H d 1 and H d 2 are assumed to not get a VEV at the electroweak scale, but instead we assume them to be very heavy and act as heavy messengers of the diagonal entries in the effective Yukawa couplings of down quarks and charged leptons, as shown in Fig. 3.For this we use the d = 3 couplings that appear in the scalar potential and couple the heavy Higgs doublets to the hyperons, see Appendix A.
After integrating out the heavy messengers, the effective Yukawa couplings for chiral fermions φℓ23 take the form .In contrast, the up-quark sector remains described by Eq. (2.19).
We can see that in this model the down-quark and charged lepton Yukawa matrices are diagonal at tree-level, therefore CKM and PMNS mixing fully originate from the up and neutrino sectors, respectively.The up-quark Yukawa couplings take a similar form as in Model 1, and we can similarly describe all the flavour structure of the SM in terms of three naturally small parameters as in Eq. (2.24), obtaining three simple NP scales above electroweak symmetry breaking as shown in Fig. 2.However, here we shall take advantage of the O(1) variation of the VL fermion masses and VEVs of the hyperons (as discussed in Section 2.1) to choose slightly different ratios as and the (1,1) entry of the effective up-Yukawa matrix is further suppressed by the fixed ratio, This choice is consistent with the spectrum of Fig. 2 up to O(1) variations, now with the addition of the two heavy Higgs doublets that live with the 23-messengers and with the 12,13-messengers, respectively.This provides the following values for the ratios involving the heavy Higgs doublets, Notice that, as in Model 1, the presence of the ratio ϕ ℓ23 /M U 12 provides a highly non-generic and predictive connection between both VEVs as per Eq.(2.25), namely Therefore, a typical benchmark for the scales of Model 2 compatible with phenomenology is We conclude this section with the approximate textures (up to O(1) coefficients) predicted for the effective Yukawa couplings by Model 2,

.45)
Since here CKM mixing originates from the up sector and the charged lepton matrix is diagonal at tree-level, the leading flavour phenomenology involves FCNCs in the up sector, unlike Model 1 where the leading phenomenology involves FCNCs in the down sector along with CLFV processes.

Neutrinos
As first discussed in [30], the tri-hypercharge symmetry imposes selection rules on the Weinberg operator that potentially complicate the explanation of large neutrino mixing.Nevertheless, this problem can be solved via minimal extensions of the traditional seesaw mechanism.With respect to [30], here we propose a different, more minimal model for the neutrino sector which does not require the addition of any extra scalar beyond those already present in Models 1 and 2.
In order to describe neutrino masses and mixing, we use the two complete singlet neutrinos ν c 1,2 and the two hyperons ϕ ℓ23 and ϕ ℓ12 already included in Table 1, along with thee vector-like neutrinos N 12 , N 13 and N 23 common to both Models 1 and 2 as shown in Tables 2 and 3.The rationale is that the vector-like neutrinos, thanks to the presence of the hyperons ϕ ℓ23 and ϕ ℓ12 , can mediate couplings between the light lepton doublets and the (very heavy) singlet neutrinos ν c 1,2 , as shown in Fig. 4, which would otherwise be forbidden by the selection rules.
However, the diagrammatic approach of Fig. 4 requires to employ the mass insertion approximation where M N 23 ≫ ⟨ϕ ℓ23 ⟩, hence typically predicting suppressed neutrino mixing in conflict with measured values [2,3].Nevertheless, the diagrams in Fig. 4 give the intuition that if M N 23 ∼ ⟨ϕ ℓ23 ⟩ (and similarly M N 12,13 ∼ ⟨ϕ ℓ12 ⟩), then large neutrino mixing can be obtained.In order to verify this, we need to disregard the mass insertion approximation, because in the regime M N 23 ∼ ⟨ϕ 23 ⟩ the EFT suggested by the diagrams in Fig. 4 is not consistent.
The way to go is to introduce formally the three messenger neutrinos N 12 , N 13 and N 23 in the theory and employ the seesaw formula.We can construct the Dirac and Majorana mass matrices in the usual way by writing all gauge invariant couplings as ) where we can take advantage of a global U (2) symmetry relating ν c 1 and ν c 2 to set M 12 = 0 without loss of generality.The full neutrino mass matrix is given as Since m D only contains couplings to the Higgs doublet H u 3 that breaks electroweak symmetry, while M N in all cases contains heavy scales in the multi-TeV region or above, we are in the regime m D ≪ M N where we can safely apply the seesaw formula.
But first we note that after exchanging the hyperons by their VEVs ⟨ϕ ℓ12 ⟩ ∼ v 12 and ⟨ϕ ℓ23 ⟩ ∼ v 23 , the submatrix in M N that corresponds to the VL messengers contains two eigenvalues at v 12 and one at v 23 (as we expect from the spectrum in Fig. 2) only for the choice of bare masses m where we have assumed O(1) coefficients x ij and v 23 ≪ v 12 as predicted from Eq. (2.25).The choice of bare masses m N ij above may seem contrary to the prescription M N 23 ∼ v 23 (and similarly ) that we anticipated from the diagrammatic approach, but we need to note that m N ij are not necessarily the physical masses of the messenger neutrinos.Indeed, in the regime where the messenger neutrinos and the VEVs of the hyperons are of the same order, the mixing terms among N ij are very relevant.After taking them into account, our choice of bare masses delivers M N 23 ∼ v 23 and M N 12,13 ∼ v 12 , as the physical masses of the messenger neutrinos.After applying the seesaw formula to extract the light neutrino masses, we verify that messenger neutrinos at these scales mediate O(1) couplings between the light lepton doublets and the singlet neutrinos.Indeed, we obtain that to leading order all the v 12 and v 23 scales cancel between the numerator and the denominator.Namely, we find If we assume M 11 ≪ M 22 as in sequential dominance [39][40][41], and x ij = z ij = 1 for simplicity, then we have This result is similar to what one would obtain in a minimal type-I seesaw extension of the SM.Indeed the texture above predicts two very light active neutrinos (and one massless) if M 11 ≈ 10 15 GeV, and is capable to explain large neutrino mixing by fitting the O(1) coefficients to existing data.Notably, this implementation of the seesaw mechanism requires no addition of extra scalars beyond those already introduced to explain the flavour structure of the charged fermion sector.Moreover, there is no need to impose small couplings nor assuming that the scales v 12 or v 23 are very heavy in order to achieve small neutrino masses (meaning that such scales can be in the multi-TeV region), as is commonly assumed in the literature.We remark that the three neutrino messengers N 12 , N 13 and N 23 obtain masses at the same scale of the Z ′ 12 and Z ′ 23 bosons to which they couple and which could live in the multi-TeV region, meaning that they could carry meaningful phenomenology if such Z ′ bosons are discovered.

Gauge boson couplings and complete fermion mixing
The phenomenology of the tri-hypercharge framework is mostly driven by the Z ′ 12 and Z ′ 23 gauge bosons that arise from the 12 and 23 breaking, respectively, and to some extent also to the Z boson, as firstly discussed in [30] where the couplings of such gauge bosons to chiral fermions can be found.
The only difference with respect to the study in [30] is the inclusion of the heavy messengers that mediate the effective Yukawa couplings in the complete models presented here.Beyond dictating the origin and size of chiral fermion mixing, which controls the size and nature of potential flavour-changing neutral currents (FCNCs) mediated by the gauge bosons, the vector-like fermion messengers also couple to the gauge bosons and mix with chiral fermions, hence potentially providing new sources of flavour-violation beyond those that originate from chiral fermion mixing.The fermion couplings of the gauge bosons may be represented as 6 × 6 matrices in flavour space, with the vector-like fermion families F 12 , F 13 and F 23 (with F = U, D, E, N for Model 1, and F = U, N only in Model 2) taken as fourth, fifth and sixth family respectively.For simplicity, we start by treating separately the couplings of chiral and vector-like fermions with the gauge bosons, in order to write in the interaction basis 6 , where f = u, d, e, ν with i, j = 1, 2, 3 and a, b = 4, 5, 6.The coupling matrices above are obtained from the covariant derivatives in [30] κ where i refers to the i-th hypercharge of the f i L,R fermion (and similarly for the vector-like fermions F ), g Y and g L are the conventional gauge couplings of SM hypercharge and SU (2) L respectively, where the gauge couplings are related by the following matching conditions The next step is to rotate the full 6 × 6 matrices containing the gauge boson couplings to the mass basis (denoted with hat notation) for each charged sector 7 , where ψ = (f, F ) generically denotes fermions with the same electric charge, without distinguishing between chiral and vector-like.The mixing matrices V ψ L,R , obtained by perturbatively diagonalising 23 , hence only the model independent observables appear in the right plot.The benchmark of equal couplings is motivated by a possible GUT origin [30], where the three gauge couplings are of the same order in the multi-TeV.
This bound is well compatible with the model-independent phenomenology of Z ′ 23 , which is dominated by the production of dilepton tails at the LHC and the modification of electroweak precision observables via mixing 9 with the SM Z boson [30,31].In particular, assuming all Yukawa couplings of the theory to be of O(1), and assuming g 3 (10 TeV) ≃ 0.6 as suggested by a GUT origin [32], we obtain the same bound M Z ′ 23 > 5 TeV from both µ → eγ and from dilepton searches at the LHC.Therefore, to be conservative we set v 23 ∼ O(10 TeV) and, thanks to the predictive relation in Eq. (2.25), this fixes the whole spectrum of both models as in Fig. 2, and in particular the mass of Z ′ 12 as M Z ′ 12 ∼ v 12 ∼ O(10 3 TeV), well compatible with all the bounds shown in Table 6.In contrast with Model 1, in Model 2 CKM mixing originates from the up sector, with the down-quark and charged lepton Yukawa couplings being diagonal at tree-level.Therefore, Z ′ 23 can only mediate FCNCs involving up-quarks.We find that the approximate U (2) 5 symmetry protects from contributions to D − D mixing, even in the presence of the VL quarks U ij .Indeed only t → c, u FCNCs can be sizable, but these are poorly bounded from the experiment, leading to no significant bounds over the parameter space in Fig. 6b beyond the model-independent ones that correspond to the production of dilepton tails at the LHC and the modification of electroweak precision observables.directly sensitive to the heavy masses of the F 23 VL fermions, and to fundamental Yukawa couplings of the model expected to be of O (1).Box diagrams of this kind have been computed previously in the literature [56][57][58].Adapting these results to our model, assuming , we obtain effective contributions to Q sd,cu As shown in Table 7, from the experimental bounds over the Q sd,cu Finally, we comment that a box diagram similar to those of Fig. 7 but involving charged leptons also arises in Model 1, contributing to µ → 3 e.However, the associated bounds turn out to be much weaker than those from meson mixing, roughly M E 23 O(1 TeV) and hence of the same order as the bounds from tree-level Z-mediated processes.

Conclusions
Tri-hypercharge is based on assigning a separate gauge hypercharge to each fermion family.This simple framework has been shown to explain successfully the origin of hierarchies in the flavour structure of the SM [30], and may arise from a gauge unified framework [32].
In this paper we have proposed and studied two minimal, ultraviolet complete, renormalisable tri-hypercharge theories of flavour.These models are the most minimal such models, in the sense of number of total degrees of freedom and representations, which can account for the quark and lepton (including neutrino) masses and mixing parameters in a natural way, namely where all observed mass and mixing hierarchies are explained in terms of ratios of high energy mass scales.These two models are similar from the low-energy point of view but differ in the heavy messenger sector that completes the theory, with consequently different flavour-changing phenomenology.
In the first model all the heavy messengers are vector-like, SU (2) singlet fermions, hence containing the simplest scalar potential, and in the second model several of the vector-like fermions are replaced by just two heavy Higgs doublets, hence being more minimal in terms of total degrees of freedoms and representations but having more terms in the scalar potential.that perform electroweak symmetry breaking couple to heavy degrees of freedom at tree-level, especially in Model 2 where we find that H d 3 couples to the heavy Higgs doublets and the hyperons.This is a consequence of the well-known hierarchy problem.Then the radiative stability of the scales in Fig. 2 requires fine-tuning of the couplings of the scalar potential to some degree, as discussed e.g. in [24] for a related framework.In order to avoid such fine-tuning one would need to impose extra dynamics like Supersymmetry or a strongly coupled sector in order to protect the masses of fundamental scalars from radiative corrections, which is beyond the scope of this paper.

B Models with less than four hyperons
In both minimal models presented in the main text, we have considered only the four hyperons . However, as first shown in [30], a simple spurion analysis reveals that one can exchange the ℓ-type hyperons by cubic powers of the q-type hyperons, i.e.
Therefore, in principle it is possible to remove ϕ ℓ12 and ϕ ℓ23 since ϕ q12 and ϕ q23 suffice to spontaneously break the tri-hypercharge symmetry in the desired way of Eq. (2.12), and also can generate all the spurions needed to populate the effective Yukawa matrices.Assuming UV completions where all the heavy messengers are vector-like fermions (like in Model 1 of the main text), the spurionic relations in Eqs.(B.68) and (B.69) lead to models that contain only the two hyperons ϕ q12 and ϕ q23 and the two Higgs doublets H u,d 3 as the only scalars of the theory, hence having the most economical scalar sector and the simplest scalar potential (and being automatically free from the appearance of physical Goldstone bosons).
Indeed by using only ϕ q12 and ϕ q23 we can write the effective Yukawa matrices in an EFT framework as where the cut-off of the EFTs are not shown, we assume that a power of Λ −1 1 accompanies every insertion of ϕ q12 and a power of Λ −1 2 accompanies every insertion of ϕ q23 .From the inspection of the matrices above, we already notice that the 23-mass hierarchy m 2 /m 3 ∼ ϕ 3 q23 /Λ 3 2 is generated at dimension-7 in the EFT while V cb ∼ ϕ q23 /Λ 2 , which is numerically close, is generated at only dimension-5.Nevertheless, after exchanging the hyperons by their VEVs, we can assume the following numerical values, to obtain the approximate textures (up to O(1) coefficients) The textures above are successful to explain the flavour structure of the SM, albeit requiring some dimensionless coefficients of O(λ) to generate the right V cb and V ub .We consider this acceptable, and perhaps could be avoided if the mass of a VL quark in the UV theory provides an extra λ factor beyond the simplified assumption of Eq. (B.73).
However, we find that in order to UV-complete the effective theory via SU (2) L singlet VL fermions, a total of 23 VL fermion representations are needed for the charged fermion sector only, as shown in Table 8, in contrast with only 9 needed in Model 1 of the main text.Moreover, some of the VL fermions need to be charged under the three family hypercharges.Hence we find that in models with only two hyperons ϕ q12 and ϕ q23 , despite containing the simplest scalar sector and the most economical low-energy theory, the UV-complete theories seem to be far from minimal.
In comparison with Model 1 of the main text, the model sketched here contains two less hyperons but at least 14 more VL fermions.This counting is obtained without exploring the neutrino sector, which will require the addition of more neutrino messengers and the full neutrino mass matrix will potentially contain various powers of the VEVs ⟨ϕ q12 ⟩ and ⟨ϕ q23 ⟩, perhaps making it more difficult to explain large neutrino mixing.Instead, if we seek for UV completions as in Model 2 of the main text, we will need to introduce 9 heavy Higgs doublets for the down and charged lepton sector   2 and 3, for vector-like fermions (highlighted in yellow), the conjugate partners are also included but not explicitly shown.
(plus 7 VL fermions for the up sector), hence being less minimal and also complicating the scalar potential.
Perhaps an alternative would be to introduce ϕ (0, 1 2 ,− 1 2 ) ℓ23 only, without introducing ϕ . This is exactly the model with 3 hyperons first discussed in Section 3.3 of [30].In this case, the effective Yukawa matrix (showing only the up sector for simplicity) reads hence now both m 2 /m 3 and V cb are generated at dimension-5 in the EFT expansion, but in contrast with the models of the main text now we generate first family masses at dimension-8 rather than dimension-6, due to the fact that ϕ ( 1 2 ,− 1 2 ,0) ℓ12 is not present but generated as ϕ ℓ12 ∼ ϕ 3 q12 .As first noted in [30], from the pure EFT point of view, this provides a nice description of the SM flavour hierarchies thanks to further hyperon insertions for the smallest (effective) Yukawa couplings.
However, we find that in order to UV-complete this effective theory via SU (2) L singlet VL fermions, a total of 15 VL fermion representations are needed for the charged fermion sector only, in contrast with only 9 needed in Model 1 of the main text.Therefore, in comparison with Model 1 of the main text, this model would contain only one less hyperon but 6 more VL fermions.On top of this, in this model the hyperons ϕ q23 and ϕ ℓ23 have a quartic coupling in the scalar potential of the form ϕ 3 q23 ϕ ℓ23 .This breaks the global phases of both hyperons.In contrast, ϕ q12 cannot couple to other scalars beyond quadratic self-conjugate terms, hence a physical Goldstone boson potentially arises associated to the unbroken phase of ϕ q12 .This problem persists in UV completions based on the addition of heavy Higgs doublets, as in Model 2 of the main text, where we find that at least 5 heavy Higgs doublets are needed for the down and charged lepton sectors.

C EFT formalism for FCNCs
The leading FCNCs in the quark sector are connected to the following operators contributing to K − K and D − D mixing Now we apply a Fierz transformation to the LR operators in order to match to the basis of Refs.[45][46][47], from which we take the bounds, where e = 4πα QED is the QED gauge coupling.

< l a t e x i t s h a 1 _
b a s e 6 4 = " r d a 6 0 F J s U 9 r 3 o m o v j B v B F v p E I c Y = " > A A A C O X i c b V D L S s N A F J 3 U V 6 2 v q k s 3 w S K 4 k J J U q C 6 L b t w o F e w D m h A m 0 9 t 2 6 G Q S Z y a F E v p b b v w L d 4 I b F 4 q 4 9 Q e c p F 3 Y 1 g M D h 3 P u Y + 7 x I 0 a l s q x X I 7 e y u r a + k d 8 s b G 3 v 7 O 4 V 9 w + a M o w F g Q Y J W S j a P p b A K I e G o o p B O x K A A 5 9 B y x 9 e p 3 5 r B E L S k D + o c Q R u g P u c 9 i j B S k t e s e 5 0 w 0 k a w b Y H B s 6 c e 8 + d u c e L G B X S N L + 0 0 t r 6 x u Z W e V v f 2 d 3 b P z A O j 3 o i j D k m X R y y k N 9 7 S B B G A 9 K V V D J y H 3 G C f I + R v j e 9 z u r 9 B 8 I F D Y O O n E X E 9 d E 4 o C O K k V T S w K g 6 t u 7 4 S E 4 w Y s l t W r N M 5 z y / c z / p k F 5 6 p j u u r g + e w C t 4 0 5 6 1 d + 1 D + 5 y 3 l r T C c w w W o H 3 / A h m H o w 0 = < / l a t e x i t > O(10 TeV) < l a t e x i t s h a 1 _ b a s e 6 4 = " L Z l u e R 1 2 8 6 3 H r y b m X H N B 9 h P a I X h P n / w c H P l j b 3 e 8 + 8 0 f 7 n / u x r E C 2 / A O R u D B H u z D V z i E A A T 8 g m u 4 h b v e V e / G A c e 5 T 3 V 6 X c 0 W / B P O 2 h 8 b Q 7 J E < / l a t e x i t > d V P d 2 t 7 Z 3 d P 2 D 3 o 8 T h n C F o p p z A Y e 5 J i S C F u C C I o H C c M w 9 C j u e 5 P r o t 5 / w I y T O L o T 0 w Q 7 h b m 5 t 7 + x a e / t N F c Y S k w Y O W S j b A V K E U U E a m m p G 2 p E k i A e M t I L h d e 6 3 R k Q q G o p 7 P Y 6 I x 1 F f 0 B 7 F S G e S b 5 2 7 H X P k J y 5 H e i B 5 c n e T p q 6 i 3 G V I 9 B m B N T 8 5 S x + S + L S b u n I i m a 5 n + l

Figure 2 :
Figure 2: Illustrative spectrum of NP scales in Models 1 and 2. Notice that the hyperons ϕ, third family Higgs doublets H u,d 3 , messenger neutrinos N and messenger up-type quarks U are common to both models.

λ 2 v
SM + h.c.,(2.33)    which can explain the flavour structure of the SM with all fundamental couplings being O(1).

Figure 3 :
Figure 3: Diagrams at the origin of light down-quarks and charged lepton masses in Model 2. The up-sector is described by diagrams such as those of Fig. 1.

Figure 4 :
Figure 4: Illustrative diagrams for the generation of effective Majorana masses for active neutrinos, where i = 1, 2.

Figure 6 :
Figure 6: Parameter space of the 23-symmetry breaking, where M Z ′23 is the mass of the Z ′ 23 gauge boson and g 3 is the gauge coupling of the U (1) Y3 gauge group.We consider respectively the non-generic fermion mixing predicted by Model 1 (left) and by Model 2 (right) where no significant FCNCs are mediated by Z ′ 23 , hence only the model independent observables appear in the right plot.The benchmark of equal couplings is motivated by a possible GUT origin[30], where the three gauge couplings are of the same order in the multi-TeV.

Figure 7 :
Figure 7: 1-loop box diagrams contributing to D − D mixing (left) and to K − K mixing (right), with VL fermions D 23 , U 23 and Higgs doublets H u,d 3 running in the loop.

1 operators
we obtain bounds over M D 23 and M U 23 of O(100 TeV), which are compatible with our expectations of M U 23 ,D 23 ≳ O(10 3 TeV) according to the spectrum of Fig. 2.

Table 2 :
Scalar and vector-like fermion content of Model 1.Notice that for vector-like fermions (highlighted in yellow), only the left-handed fields U, D, E, N , with the same electric charges as the chiral fermions u c , d c , e c , ν c in Table

Table 3 :
Scalar and vector-like fermion content of Model 2. As in Table2, for vector-like fermions (highlighted in yellow), the conjugate partners are also included but not explicitly shown.

Table 4 :
Approximate charged fermion mixing in Model 1.The angles not shown are zero at tree-level.We also neglect up-quark mixing, assuming that the CKM mostly originates from the down sector as the textures in Eqs.(2.13) and (2.14) suggest.
the conventional weak mixing angle and the weak-like mixing angles θ W

Table 8 :
Scalar and vector-like content of the renormalisable model with two hyperons (charged fermion sector only).As in Tables