Tri-hypercharge: a separate gauged weak hypercharge for each fermion family as the origin of flavour

We propose a tri-hypercharge (TH) embedding of the Standard Model (SM) in which a separate gauged weak hypercharge is associated with each fermion family. In this way, every quark and lepton multiplet carries unique gauge quantum numbers under the extended gauge group, providing the starting point for a theory of flavour. If the Higgs doublets only carry third family hypercharge, then only third family renormalisable Yukawa couplings are allowed. However, non-renormalisable Yukawa couplings may be induced by the high scale Higgs fields (hyperons) which break the three hypercharges down to the SM hypercharge, providing an explanation for fermion mass hierarchies and the smallness of CKM quark mixing. Following a similar methodology, we study the origin of neutrino masses and mixing in this model. Due to the TH gauge symmetry, the implementation of a seesaw mechanism naturally leads to a low scale seesaw, where the right-handed neutrinos in the model may be as light as the TeV scale. We present simple examples of hyperon fields which can reproduce all quark and lepton (including neutrino) masses and mixing. After a preliminary phenomenological study, we conclude that one of the massive $Z'$ bosons can be as light as a few TeV, with implications for flavour-violating observables, LHC physics and electroweak precision observables.


Introduction
The existence of the three families of quarks and leptons is one of the fundamental mysteries of the Standard Model (SM).Indeed many of the parameters of the SM are associated with the resulting quark and lepton mass patterns and mixings.The low energy masses of quarks and charged leptons may be expressed approximately 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 The hierarchical patterns of masses and CKM mixing, and the possibility that they might be understood in the form of a more fundamental theory of flavour beyond the SM, has classically been denoted as the flavour puzzle.The discovery of neutrino oscillations, which proves that at least two neutrinos have non-zero mass, has made the flavour puzzle difficult to ignore, enlarging the flavour sector with extra neutrino mixing angles [2,3] tan plus very tiny neutrino masses which follow either a normal or inverted ordering.The need for a theory of flavour to understand all fermion masses and mixings is now more urgent than ever.Theories of flavour may involve new symmeties structures (global, local, continuous, discrete, abelian, non-abelian...) beyond the SM group, possibly broken at some high scale down to the SM.Traditionally, new gauge structures beyond the SM have been considered to be flavour universal, as grand unified theories (GUTs) which usually embed all three families in an identical way, or even extended GUTs which embed all three families in a single representation (usually along with extra exotic fermions).Alternatively, there exists the well-motivated case of family symmetries which commute with the SM gauge group, and are then spontaneously broken, leading to family structure.However, there are other less explored ways in which the SM gauge group could be embedded into a larger gauge structure in a flavour non-universal way.In particular, the family decomposition of the SM gauge group (including a hierarchical symmetry breaking pattern down to the SM) was first proposed during the 80s and 90s, with the purpose of motivating lepton nonuniversality [4][5][6][7] or assisting technicolor model building [8][9][10].However, the natural origin of flavour hierarchies in such a framework was not explored until more recently in [11][12][13].Here it was proposed that the flavour non-universality of Yukawa couplings in the SM might well find its origin in a flavour non-universal gauge sector, broken in a hierarchical way down to the SM.Interestingly, model building in this direction has received particular attention in recent years [14][15][16][17][18][19].With the exception of Ref. [19], the remaining recent attempts have been motivated by the need to obtain a TeV scale vector leptoquark from Pati-Salam unification in order to address the so-called Banomalies 1 .Therefore, all these setups share a similar feature: a low scale SU (4) gauge group under which only the third family of SM fermions transforms in a non-trivial way.In contrast, in this work we want to explore the capabilities of flavour non-universality to address the flavour puzzle in a more minimal, simple and bottom-up approach.
We propose that the SM symmetry originates from a larger gauge group in the UV that contains three separate weak hypercharge gauge factors, which we will denote as the tri-hypercharge (TH) U (1)3 Y gauge group.We will associate each of the three hypercharge gauge groups with a separate SM family, such that each fermion family i only carries hypercharge under the corresponding U (1) Y i factor.This ensures that each family transforms differently under the gauge group U (1) 3  Y , which avoids the family repetition of the SM, and provides the starting point for a theory of flavour.For example, assuming that a single Higgs doublet only carries third family hypercharge, then only the third family Yukawa couplings are allowed at renormalisable level.With two Higgs doublets carrying third family hypercharge, we show that the naturalness of the scheme increases.This simple and economical framework naturally explains the heaviness of the third family, the smallness of V cb and V ub , and delivers Yukawa couplings that preserve an accidental and global U (2) 5 flavour symmetry acting on the light families, which is known to provide a good first order description of the SM spectrum plus an efficient suppression of flavour-violating effects for new physics [25].Remarkably, this appears to be the simplest way to provide the U (2) 5 flavour symmetry 2 .The masses of first and second family fermions, along with the CKM mixing, then appear as small breaking sources of U (2) 5 that arise after the cascade spontaneous symmetry breaking of U (1) 3  Y down to SM hypercharge, which can be parameterised in a model-independent way in terms of spurions.In a realistic model, the spurions will be realised by a choice of "hyperon" scalars which transform under the different family hypercharge groups, breaking the tri-hypercharge symmetry.We will motivate a specific symmetry breaking chain where dynamics at a low scale, which could be as low as the TeV, explain the flavour hierarchies m 2 /m 3 , while dynamics at a heavier scale explain m 1 /m 2 .This symmetry breaking pattern will sequentially recover the approximate flavour symmetry of the SM, and provide a natural suppression of FCNCs for TeV new physics, while the rest of flavour-violating effects are suppressed by a naturally heavier scale.
The paper is organised as follows.In Section 2 we introduce the TH gauge group, along with the fermion and Higgs doublet content of the model.We discuss the implications for third family fermion masses along with the mass hierarchy between the top and bottom/tau fermions.In Section 3 we study the origin of charged fermion masses and mixing in the TH model, firstly via a spurion formalism which reveals model-independent considerations, and secondly by introducing example models with hyperons.In Section 4 we study the origin of neutrino masses and mixing in the TH model.In particular, we discuss the impact of the U (2) 5 flavour symmetry over the dimension-5 Weinberg operator, and afterwards we provide an example type I seesaw model where neutrino masses and mixing can be accommodated.In Section 5 we perform a preliminary exploration of the phenomenological implications and discovery prospects of the U (1) 3  Y theory of flavour.Finally, Section 6 outlines our main conclusions.

Tri-hypercharge gauge theory
The tri-hypercharge gauge group is based on assigning a separate gauged weak hypercharge to each fermion family, in such a way that the ith 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 SM weak hypercharge.Anomalies cancel separately for each family, as in the SM, but without family replication.The TH gauge group is broken down to the SM via appropriate SM singlet scalars, which however carry family hypercharges.We denote these fields linking the family hypercharges as hyperons.The TH group could be broken down to the SM in different ways, however we motivate the following symmetry breaking pattern, This choice is well supported by symmetry arguments that will have phenomenological consequences: At high energies, the TH group discriminates between the three SM fermion families, explicitly breaking the approximate U (3) 5 flavour symmetry of the SM.At a heavy scale v 12 , the first and second hypercharges are broken down to their diagonal subgroup, and the associated potentially mediates dangerous 1-2 flavour-changing neutral currents (FCNCs).Nevertheless, the gauge group below the scale v 12 preserves an accidental U (2) 5 flavour symmetry.The groups are broken down to their diagonal subgroup (SM hypercharge) at a scale v 23 , and the associated Z ′ boson is protected from mediating the most dangerous FCNCs thanks to the U (2) 5 symmetry.In this manner, the most dangerous FCNCs are suppressed by the heavier scale v 12 , while the scale v 23 can be very low with interesting phenomenological implications.We will see that dynamics connected to the scale v 12 will play a role in the origin of the family hierarchy m 1 /m 2 , while dynamics connected to the scale v 23 will play a role in the origin of m 2 /m 3 .The distribution of the various scales and the approximate flavour symmetries that apply at each scale are summarised in the diagram of Fig. 1.
Provided that the SM Higgs only carries third family hypercharge, H(1, 2) (0,0,− 1 2 ) , then the third family Yukawa couplings are allowed at renormalisable level and an accidental U (2) 5 flavour symmetry acting on the light families emerges in the Yukawa sector, where H is the CP -conjugate of H.This setup already provides an explanation for the smallness of light fermion masses with respect to the third family, along with the smallness of quark mixing, as they all must arise from non-renormalisable operators which minimally break the U (2) 5 symmetry.
Although this is a good first order description of the SM spectrum, the question of why the bottom and tau fermions are much lighter than the top remains unanswered, and assuming only a single Higgs doublet, a tuning of order 2% for the bottom coupling and of 1% for the tau coupling would be required.Given that m s,µ ∝ λ 5 m t while m c ∝ λ 3 m t , this setup also requires to generate a stronger fermion hierarchy in the down and charged lepton sectors with respect to the up sector, unless the tuning in the bottom and tau couplings is extended to the second family.As we shall see shortly, the U (1) 3 Y model (and very likely a more general set of theories of flavour based on the family decomposition of the SM group) predicts a similar mass hierarchy for all charged sectors.Due to the above considerations, it seems natural to consider a type II two Higgs doublet model (2HDM), where both Higgs doublets only carry third family hypercharge, where as usual for a type II 2HDM, FCNCs can be forbidden by e.g. a Z 2 discrete symmetry or by supersymmetry (not necessarily low scale), which we however do not specify in order to preserve the bottom-up spirit of this work.In any case, tan β = v u /v d ∼ λ −2 ≈ 20, which is compatible with current data (see e.g.[28,29]), will provide the hierarchy between the top and bottom/tau masses with all dimensionless couplings being O(1).Such an overall hierarchy between the down and charged leptons sector with respect to the up sector is extended to all families, providing a better description of second family charged fermion masses as we shall see.
3 Charged fermion masses and mixing

Lessons from the spurion formalism
In all generality, we introduce U (2) 5 -breaking spurions Φ in the Yukawa matrices of charged fermions where each spurion carries non-trivial charge assignments under U (1) 3 Y .In an effective field theory (EFT) approach, each spurion above can be matched to specific ratios of hyperons ϕ i over EFT cut-off scales Λ i , i.e.
where we have suppressed dimensionless couplings.The choice of hyperons and Λ i above carries all the model dependence.
Assuming that the cut-off scales Λ i of the EFT are universal, i.e. that all Λ i are common to all charged sectors, then the spurion formalism reveals some general considerations about the origin of charged fermion masses and mixing: • The same spurions (up to conjugation) appear in the diagonal entries of all matrices.Therefore, unless texture zeros are introduced in specific models, this means that the masses of second family fermions are likely to be degenerate up to dimensionless couplings, and the same discussion applies to first family fermions.This motivates again the addition of the second Higgs doublet or an alternative mechanism in order to generate the hierarchy between the charm mass and the lighter strange and muon masses.
• The same spurions appear in the (2,3) entries of the up and down Yukawa matrices.Therefore, the 2-3 mixing in both the up and down sectors is expected to be of a similar size, giving no predictions about the alignment of the CKM element V cb .The similar argument applies to 1-3 mixing and V ub .
• The spurions in the (1,2) entry of the up and down matrices are different.Therefore, specific models have the potential to give predictions about the alignment of the CKM element V us .
• The same spurion (up to conjugation) that enters in all (2,2) entries also populates the (2,3) entry of the charged lepton Yukawa matrix.Similarly, the same spurion (up to conjugation) that enters in all (1,1) entries also populates the (1,3) entry of the charged lepton Yukawa matrix.In general, this predicts left-handed µ − τ (e − τ ) mixing of O(m 2 /m 3 ) (O(m 1 /m 3 )), unless texture zeros are introduced in specific models (see Section 3.2).This leads to a sizable enhancement of LFV τ → µ and τ → e transitions above the SM predictions, mediated by heavy Z ′ bosons in the model (see Section 5).
• The spurions in the lower off-diagonal entries of the Yukawa matrices all carry independent charge assignments, so right-handed fermion mixing is model-dependent and can be different in all charged sectors.
In the following, we go beyond the spurion formalism and introduce different sets of hyperons.As we shall wee, the hyperons will provide small U (2) 5 -breaking effects via non-renormalisable operators, leading to the masses of first and second family charged fermions, along with CKM mixing.In the next few subsections, we will describe example scenarios which provide a good description of charged fermion masses and mixing.

From spurions to hyperons
The physical origin of the spurions of the previous subsection will correspond to new Higgs scalar fields that break the U (1)3 Y symmetry, which we call hyperons.The hyperons induce small U (2) 5 breaking effects at the non-renormalisable level that will lead to the masses and mixings of charged fermions.As the most straightforward scenario, we could promote the spurions in the diagonal entries of the matrices in Eq. (3.11) to hyperons, along with the off-diagonal spurions in the upper half of the down matrix 3 .In an EFT approach, the set of hyperons that we have assumed generates the following Yukawa matrices, where the universal scale Λ is the high cut-off scale of the EFT, and we ignore the O(1) dimensionless couplings of each entry.Although we have chosen only the specific set of hyperons shown, leaving some zeros in the matrices, these zeros may be filled in by higher order operators with dimension larger than 5, which so far we are ignoring.
When the hyperons develop VEVs, assumed to be smaller than the cut-off scale Λ, then each entry of the matrix will receive a suppressed numerical effective coupling given by ratios of the form ⟨ϕ⟩ /Λ, whose values can be assumed arbitrarily.Having the freedom to choose arbitrary VEVs for each hyperon, the Yukawa matrices above could provide a good first order description of charged fermion masses and CKM mixing.We choose to fix the ⟨ϕ⟩ /Λ ratios in terms of powers of the Wolfenstein parameter λ ≃ 0.224, obtaining As anticipated from the spurion formalism, the alignment of V cb and V ub is not predicted by the model.In contrast, the model predicts a relevant left-handed µ − τ (e − τ ) mixing connected to the same hyperon that provides the second family (first family) effective Yukawa couplings.Thanks to the addition of the second Higgs doublet, the model successfully explains third and second family fermion masses with O(1) dimensionless couplings.The down-quark and electron masses are also reasonably explained, although the up-quark mass is naively a factor O(λ 1.5 ) larger than current data.Notice that so far we are only assuming one universal cut-off scale Λ, while in realistic models several cut-off scales Λ may be associated to different messengers in the UV theory, which could provide a larger suppression for the up-quark effective coupling.Therefore, within the limitations of our EFT approach, the description of charged fermion masses given by the set of Eqs.(3.16-3.18) is very successful.
As an alternative example, one could also consider a model where the (1,1) spurion in Eq. (3.11) is not promoted to hyperon, but instead all the spurions in the (1,2) and (2,1) entries are promoted, so that the Yukawa matrices show an exact texture zero in the (1,1) entry, where we have omitted the high cut-off Λ suppressing each dimension-5 operator above.The VEV over Λ ratios of the new hyperons can be fixed by the requirement of addressing first family fermion masses, obtaining Yukawa matrices with texture zeros given by which provide an even better description of first family fermion masses than the original simplified model.Notice that in this scenario, a sizable left-handed e − τ mixing is no longer predicted.We conclude that the most straightforward choices of hyperons, motivated by the spurion formalism, already provide a good description of charged fermion masses and mixings.However, these simplified models leave some questions unanswered.Given that we are assuming the symmetry breaking highlighted in Eq. (2.8) and Fig. 1, we notice that there are no hyperons breaking the first and second hypercharges down to their diagonal subgroup, and we would expect those to play a role in the origin of fermion hierarchies and mixing.Moreover, in the simplified models introduced so far, several hyperons display unexplained large hierarchies of VEVs whose values are assumed a posteriori to fit the fermion masses.Given that all these hyperons participate in the 23-breaking step of Eq. (2.8), we would expect all of them to develop VEVs at a similar scale, rather than the hierarchical scales assumed.This motivates further model building.In the following subsections we discuss a couple of example models which address these issues.

Model 1: Minimal case with three hyperons
We introduce here the following set of three hyperons, Following the EFT approach of the previous subsection, we now analyse the effective Yukawa matrices obtained by combining the SM charged fermions, the Higgs doublets and the hyperons, in a tower of non-renormalisable operators preserving the U (1) 3 Y gauge symmetry, where the powers of Λ in the denominator and the dimensionless couplings of each entry are not shown.Once the hyperons above develop VEVs, we obtain very economical and efficient Yukawa textures for modeling the observed pattern of SM Yukawa couplings.In particular, the masses of second family fermions arise at dimension 5 in the EFT, while first family masses have an extra suppression as they arise from dimension-8 operators.Regarding CKM mixing, 2-3 quark mixing leading to V cb arises from dimension-5 operators, while V ub has an extra mild suppression as it arises from dimension-6 operators.In all cases, right-handed fermion mixing is suppressed with respect to left-handed mixing.This is a highly desirable feature, given the strong phenomenological constraints on right-handed flavour-changing currents [30,31], which may be mediated by heavy Z ′ bosons arising from the symmetry breaking of U (1) 3 Y (see Section 5).In good approximation, quark mixing leading to V us arises as the ratio of the (1,2) and (2,2) entries of the quark matrices above, therefore we expect where λ = sin θ C ≃ 0.224.In a similar manner, we can fix the ratio ⟨ϕ q23 ⟩ /Λ by reproducing the observed Given that both ⟨ϕ q23 ⟩ and ⟨ϕ ℓ23 ⟩ play a role in the last step of the symmetry breaking cascade (see Fig. 1), it is expected that both VEVs live at a similar scale, although they are not expected to be degenerate but differ by an O(1) factor.This way, we are free to choose which, given that ⟨H d ⟩ provides an extra suppression of O(λ 2 ) for down-quarks and charged lepton Yukawas, allows to predict all second family masses with O(1) dimensionless couplings.In contrast with the simplified models of Section 3.2, this model provides all the 23-breaking VEVs at the same scale, plus a larger 12-breaking VEV, following a mild hierarchy given by v 23 /v 12 ∼ λ.This way, the symmetry breaking of the U (1) 3 Y gauge group proceeds just like in Eq. (2.8) and Fig. 1, as desired.Having fixed all the hyperon VEVs with respect to Λ, now we are able to write the full mass matrices for each sector in terms of the Wolfenstein parameter λ, We can see that this setup provides a reasonable description of charged fermion masses and mixing.
Although the up and down-quark masses are slightly off by O(λ) factors, we remember that we are only assuming one universal cut-off scale Λ, while in realistic models several scales Λ may be associated to different messengers in the UV theory, further improving the fit of first family quark masses, as discussed in Section 3.2.All things considered, the description of fermion masses seems very efficient, considering the limitations of our EFT framework.However, the model does not predict the alignment of V us .Moreover, we also notice that right-handed s − d mixing is just mildly suppressed as s d R 12 ≃ O(λ 2 ) in this model.Given the stringent bounds over left-right scalar operators contributing to K 0 − K0 meson mixing [30,31] (which might arise in this kind of models as we shall see in Section 5), the scale v 12 can be pushed far above the TeV if V us originates from the down sector.From a phenomenological point of view, it would be interesting to find models which give clear predictions about the alignment of V us , and ideally provide a more efficient suppression of right-handed quark mixing.We shall see in the next subsection that this can be achieved by minimally extending the set of hyperons of this model.

Model 2:
Five hyperons for a more predictive setup Model 1 proposed in the previous subsection, despite its simplicity and minimality, does not give clear predictions about the alignment of the CKM matrix.Heavy Z ′ bosons arising from the symmetry breaking of U (1) 3  Y have the potential to mediate contributions to K 0 − K0 meson mixing, which could set a lower bound over the scale of U (1) 3 Y -breaking, but such contributions depend on the alignment of V us .Moreover, the largest contributions to K 0 − K0 mixing depend both on the alignment of V us and on right-handed s − d mixing, which is just mildly suppressed in Model 1.Therefore, we propose here a similar model with slightly extended hyperon content that can account for a clear prediction about the alignment of V us , plus a more efficient suppression of right-handed fermion mixing.We consider here the hyperons With this set of hyperons, the effective Yukawa couplings in the EFT are (suppressing as usual powers of Λ and dimensionless couplings) Following the same approach as with Model 1, we assign the following powers of λ to the VEV over Λ ratios in order to reproduce fermion masses and CKM mixing, Although it would seem that in this scenario there exists a mild hierarchy between 23-breaking VEVs of O(λ 2 ), since ϕ d12 only appears in the 12 entry of the down matrix, it would be very reasonable that the dimensionless coupling in that entry provides a factor λ suppression, such that all 23-breaking VEVs live at the same scale.The largest VEV is still the 12-breaking one, which is now associated to the hyperon ϕ e12 , and the mild hierarchy between scales remains as v 12 /v 23 ≃ λ.
With these assignments of VEVs over Λ ratios, the Yukawa textures are given by Just like in Model 1, this model provides a compelling description of all charged fermion masses and mixing.Notice that this scenario provides a very efficient suppression of right-handed fermion mixing.Moreover, it is clear that here V us mixing originates from the down sector, providing a more predictive setup than Model 1, which will be useful for phenomenological purposes as discussed in Section 5.
Finally we comment that the Higgs doublets and hyperons can mediate FCNCs such as K 0 − K0 mixing.In the present model, this may arise from tree-level exchange of ϕ d12 hyperons which can mediate down-strange transitions.Such a coupling originates from Q 1 ϕ d12 d c 2 H d /Λ which will lead to a suppressed down-strange coupling of order λ 2 v SM /Λ ≈ 10 −5 (taking Λ ≈ 100 TeV as expected if v 23 ≈ O(TeV)).Assuming the mass of the hyperons to be at the scale v 23 , then we expect these FCNCs to be under control.Since we assume a type II 2HDM, tree-level Higgs doublets exchange contributions are forbidden, and FCNCs mediated by the Higgs doublets can only proceed via their mixing with hyperons, carrying therefore an extra suppression via the mixing angle, along with the suppression of hyperon couplings already discussed.In more general models of this kind, such contributions to K 0 − K0 mixing could be even further suppressed depending on the order of the operator and the alignment of V us , and in all generality we expect all hyperon couplings to be suppressed by at least a factor v SM /Λ ≈ 10 −3 .A more detailed study of the phenomenology of the scalar sector in this general class of models is beyond the scope of this work.

General considerations and spurion formalism
The origin of neutrino masses and mixing requires a dedicated analysis due to their particular properties.We start by introducing U (2) 5 -breaking spurions (carrying inverse of mass dimension) for the Weinberg operator which reveals that, as expected, the U (2) 5 approximate symmetry is naively present in the neutrino sector as well.As a consequence, one generally expects one neutrino to be much heavier than the others, displaying tiny mixing with the other neutrino flavours.In the spirit of the type I seesaw mechanism, one could think of adding a U (1) 3 Y singlet neutrino as N (0, 0, 0).Such a singlet neutrino can only couple to the third family active neutrino at renormalisable level, i.e.L N ⊃ L 3 H u N + m N N N , where all fermion fields are written in a left-handed 2-component convention.This way, the coupling L 2 H u N , which is required for large atmospheric neutrino mixing, can only arise at the non-renormalisable level.Therefore, it is expected to be suppressed with respect to L 3 H u N .This seems to be inconsistent with large atmospheric neutrino mixing, at least within the validity of our EFT framework.As anticipated before, this is a consequence of the accidental U (2) 5 flavour symmetry delivered by the TH model.
Given such general considerations, we conclude that in order to obtain neutrino masses and mixing from the type I seesaw mechanism, it is required to add SM singlet neutrinos that carry tri-hypercharges (but whose hypercharges add up to zero).These neutrinos will allow to introduce U (2) 5 -breaking operators similar for all neutrino flavours, providing a mechanism to obtain the adequate neutrino mixing in a natural way.In order to cancel gauge anomalies, the most simple option is that these singlet neutrinos are vector-like.As a consequence, they can obtain their mass from the unspecified vector-like mass terms and from the VEVs of hyperons in the specific model.
Remarkably, if a given SM singlet neutrino N atm carrying non-trivial hypercharges provides the couplings L Natm ⊃ L 3 H u N atm + L 2 H u N atm , as required to explain atmospheric mixing, then the conjugate neutrino will always couple to L 3 as L 3 H u N atm , but not necessarily to L 2 .As shown in Appendix A, when the terms L 3 H u N atm enter the seesaw mechanism, they lead to a hierarchical effective neutrino mass matrix proportional to the vector-like mass terms, while the terms involving SM singlet neutrinos like N atm lead to a neutrino mass matrix where all entries are O(1) and proportional to the VEVs of the hyperons.Therefore, the simplest way to explain the observed pattern of large neutrino mixing requires that the vector-like masses are of similar or smaller order than the VEVs of the hyperons, leading to a low scale seesaw mechanism if the VEVs of the hyperons are not very large (which is consistent with current data as discussed in Section 5).

Example of successful neutrino mixing from the seesaw mechanism
In the following, we provide an example scenario which reproduces the observed pattern of neutrino mixing, as a proof of principle.According to the discussion in the previous subsection, in order to implement a type I seesaw mechanism that delivers large neutrino mixing, we need to add vector-like neutrinos that carry tri-hypercharges (but whose hypercharges add up to zero).We also need to introduce hyperons that will provide small Dirac mass terms for the active neutrinos in the form of non-renormalisable operators.Under these considerations, we start by adding the following vector-like neutrino 4 and hyperon where the charge assignments are chosen to provide large atmospheric neutrino mixing.This way, we can write the following non-renormalisable operators along with the Majorana and vector-like masses of N atm , where we have ignored the O(1) dimensionless couplings, and the hyperon ϕ is already present in both Model 1 and Model 2 for the charged fermion sector.In a similar spirit, we introduce another vector-like neutrino and other hyperons in order to obtain large solar neutrino mixing which provide the following non-renormalisable operators and mass terms, where we have ignored again the O(1) dimensionless couplings, and the hyperon ϕ is already present in Model 2 for the charged fermion sector.The hyperon ϕ ν13 (which will eventually populate the (1,3) entry of the effective neutrino mass matrix) is not required to obtain non-zero reactor mixing, which would already arise from the other operators, but it is required in order to have enough free parameters to fit all observed neutrino mixing angles and mass splittings.
Notice that the vector-like neutrinos N atm and N sol get contributions to their masses from the VEVs of the hyperons ϕ ℓ23 and ϕ sol , respectively, which we denote generically as v 23 since they both take part in the 23-breaking step of Eq. (2.8).In addition, N atm and N sol get contributions to their masses from the unspecified vector-like mass terms, that we generically denote as M VL .As shown in Appendix A (see Eq. (A.61)), given that the conjugate neutrinos N atm and N sol only couple to the third family, the seesaw formula reveals that the effective neutrino mass matrix m ν receives two main contributions: • A contribution proportional to v 23 which populates all entries of m ν with O(1) terms.
• A contribution proportional to the vector-like masses M VL , which populates only the third row and column entries of m ν with O(1) terms and the others are zero.
Therefore, if M VL ≫ v 23 , then after the seesaw mechanism the light effective Majorana neutrino mass matrix m ν will have only the third row and column being non-zero (to good approximation), which is inconsistent with the observed pattern of neutrino mixing and mass splittings.Instead, if M VL v 23 , then the contribution proportional v 23 dominates the seesaw mechanism.In this case, the conjugate neutrinos N atm and N sol become irrelevant for the seesaw mechanism, and m ν can be obtained by considering only the presence of the SM singlet neutrinos N atm and N sol and applying the seesaw formula.We construct the Dirac and Majorana matrices (ignoring O(1) dimensionless couplings) as We could have included the U (1) 3 Y singlet neutrino N (0, 0, 0), but as discussed in the previous subsection, its contributions to the Weinberg operator may be suppressed by its large Majorana mass m N , resulting in possibly negligible contributions to the seesaw mechanism.We are therefore free to assume that such a neutrino N (0, 0, 0), if exists, is in any case decoupled from the seesaw, while the atmospheric and solar SM singlet neutrinos N atm and N sol could yield dominant and subdominant contributions, resulting in a natural normal neutrino mass hierarchy as in sequential dominance [32][33][34].
More generally, after applying the seesaw formula using Eq. ( 4.48), we obtain the light effective Majorana neutrino mass matrix (ignoring again O(1) dimensionless couplings) as Given the symmetry breaking pattern of the model shown in Eq. (2.8), we take ⟨ϕ e12 ⟩ ≃ O(v 12 ) and 5 .Motivated by our discussion of the charged fermion sector (see Section 3), we consider the relation v 23 /v 12 ≃ λ.By inserting such VEVs, we obtain

.50)
If Λ sol = Λ atm , we observe that there exists a mild hierarchy of order λ 2 between the 12 and 23 sectors in the matrix above.Considering the dimensionless coefficients that we have ignored so far, the numerical diagonalisation of m ν would require some parameters of O(0.01) in order to explain the observed neutrino mixing angles and mass splittings [2,3].The situation can be improved if we assume a mild hierarchy between cut-off scales Λ atm /Λ sol ≃ λ, obtaining to leading order for each entry (ignoring dimensionless coefficients), where we have introduced the SM VEV as ⟨H u ⟩ = v SM (ignoring the factor 1/ √ 2).Considering now the dimensionless coefficients in the matrix above, we find that numerical diagonalisation can accommodate all the observed neutrino mixing angles and mass splittings [2,3] with O(1) parameters, and we are able to reproduce both normal and inverted ordered scenarios.Notice that we have been driven to a scenario where the vector-like neutrinos get Majorana masses from the VEVs of hyperons in the model.Furthermore, the vector-like masses necessarily have to be of the same or smaller order than the VEVs of the hyperons in order to explain the observed pattern of neutrino mixing.Therefore, in the particular example included in this section, the vector-like neutrinos get a mass at the scale v 23 of the 23-breaking step in Fig. (1), which could happen at a relatively low scale as we shall see in Section 5.As a consequence, the vector-like neutrinos involved in the seesaw mechanism are expected to be relatively light, and the high energy cut-offs of the EFT Λ atm and Λ sol are expected to provide most the suppression for tiny neutrino masses.As anticipated before, due to the U (2) 5 flavour symmetry provided by the TH model, we have been driven to a low scale seesaw in order to predict the observed pattern of neutrino mixing.

Couplings of the heavy Z ′ bosons to fermions In Sections 3 and 4 we have discussed examples of U (1) 3
Y models which provide a compelling description of all fermion masses and mixings, and we have highlighted model-independent features which are intrinsic to the U (1) 3  Y framework.Under well-motivated arguments, we have assumed that the symmetry breaking pattern of the U (1) 3  Y group down to the SM is described by Eq. (2.8) and Fig. 1, in such a way that at a high scale v 12 , the group U (1) Y 1 × U (1) Y 2 is broken down to its diagonal subgroup.The remaining group U (1) Y 1 +Y 2 × U (1) Y 3 is broken down to SM hypercharge at a lower scale v 23 .The hierarchy between the scales v 12 and v 23 generally plays a role on the origin of flavour hierarchies in the SM, although in specific models we have found that a mild hierarchy A massive gauge boson Z ′ 12 is predicted to live at the higher scale v 12 , displaying intrinsically flavour non-universal couplings to the first two families of SM fermions.Similarly, another massive boson Z ′ 23 lives at the lower scale v 23 .The pattern of symmetry breaking is such that Z ′ 23 has flavour universal couplings to first and second family fermions, while the couplings to the third family are intrinsically different.In the following, we include the coupling matrices in family space (from the covariant derivatives in Appendices B and C), ignoring fermion mass mixing, , (5.52) where Y ψ L,R is the SM hypercharge of ψ L,R 6 , where ψ is a 3-component column vector containing the three families ψ = u i , d i , e i , ν i .Explicitly, , respectively, ignoring couplings to the SM singlet neutrinos discussed in the previous section 7 .
Including fermion mass mixing, we would have ψ L,R = V ψ L,R ψL,R with ψL,R containing the mass eigenstates and V ψ L,R being the non-generic mixing matrices obtained after diagonalising the Yukawa matrices for a given model.Notice that the couplings to right-handed fermions are larger since their hypercharges are generally larger in magnitude than those of left-handed fermions.The SM hypercharge gauge coupling g Y (M Z ) ≃ 0.36 is entangled to the g i couplings via the relations (5.54) The expressions above reveal a lower bound on the gauge couplings g i g Y .Throughout this work we have considered a bottom-up approach where the U (1) 3  Y model is just the next step in our understanding of Nature, which reveals information about the origin of flavour, but nevertheless is an EFT remnant of a more fundamental UV-complete theory.In this spirit, we have studied the RGE evolution of the gauge couplings g i , obtaining that for g i (TeV) ≃ 1 the model can be extrapolated to the Planck scale (and beyond).Instead, for g i (TeV) ≃ 2, a Landau pole is found at a scale O(10 4 TeV), which anyway seems like a reasonable scale for an UV embedding, given that we expect the cut-off scale of the effective Yukawa operators (see Section 3) to be around O(10 2 TeV) in order to provide the required suppression for charged fermion masses.Therefore, in order to protect the perturbativity of the model, we avoid considering g i > 2 in the phenomenological analysis.Nevertheless, we highlight a natural scenario where the three gauge couplings have a similar size g 1 ≃ g 2 ≃ g 3 ≃ √ 3g Y , which could be connected to a possible gauge unification.This benchmark is depicted as a dashed horizontal line in Figs. 2 and 3.

The high scale boson Z ′ 12
In any implementation of the U (1) 3 Y model, Z ′ 12 is expected to mediate sizable tree-level transitions between first and second generation left-handed quarks, either in the up or down sector depending on the alignment of the CKM matrix predicted by the specific model.Furthermore, our analysis in Section 3 reveals that U (1) 3  Y models generally predict non-vanishing charged lepton mixing and mixing among right-handed quarks.This way, contributions to K 0 − K0 and D 0 − D0 meson mixing [30,31], along with CLFV processes such as µ → eγ [1], have the potential to push the scale v 12 far above the TeV.
Being more specific, for Model 1 described in Section 3.3 we find the stringent bounds over v 12 to come from the scalar and coloured operator ( 12 at tree-level (and applying a Fierz rearrangement), which contributes to K 0 − K0 mixing.Model 1 predicts the up and down left-handed mixings to be similar up to dimensionless couplings, which must therefore play some role in the alignment of the CKM matrix.In either case, mildly suppressed right-handed s − d mixing s d R 12 ≃ O(λ 2 ) is predicted.If V us originates mostly from

The low scale boson Z ′ 23
Given that the high scale symmetry breaking can be as low as 10-20 TeV for specific models, and considering the hierarchy of scales v 23 /v 12 ≃ λ suggested by these specific models in Section 3, it is possible to find the low scale breaking v 23 near the TeV.Since Z ′ 23 features flavour universal couplings to the first and second families, the stringent bounds from K 0 − K0 mixing and µ → eγ are avoided, in the spirit of the GIM mechanism.This way, Z ′  23 can live at the TeV scale, within the reach of the LHC and future colliders.
Any implementation of the U (1) 3 Y model predicts small mixing between Z ′ 23 and the SM Z boson given by the mixing angle (see Appendix C) where M 0 Z and M 0 Z ′

23
are the masses of the Z and Z ′ 23 bosons in the absence of mixing, respectively, and g L is the gauge coupling of SU (2) L .This mixing leads to a small shift on the mass of the Z boson, which has an impact on the so-called ρ parameter which is predicted as ρ = 1 at tree-level in the SM.This is a consequence of custodial symmetry in the Higgs potential, which is explicitly broken in our model via Z − Z ′ 23 mixing leading to deviations in ρ.The fact that in our model M Z is always shifted to smaller values leads to ρ > 1 at tree-level.Given that M Z is commonly an input experimental parameter of the SM used in the determination of g Y and g L , the downward shift of M Z with respect to the SM prediction would be seen from the experimental point of view as an upward shift of M W with respect to the SM prediction.Nevertheless, the experimental picture of M W is puzzling after the recent measurement by CDF [35].This measurement points towards M W being larger than the SM prediction with high significance, but it is in tension with the combination of measurements by LHC, LEP and Tevatron D0 [1].Neglecting the recent CDF measurement for the moment, current data9 provides ρ = 1.0003 ± 0.0005 [1] (assuming that both the oblique parameters T and S are non-zero, as we expect in our model).We obtain the approximate bound g 3 /M Z ′ < 3.1 TeV at 95% CL, which translates to an approximate bound over the mixing angle of sinθ Z−Z ′ 23 < 0.001.Z − Z ′ 23 mixing also shifts the couplings of the Z boson to fermions, leading to an important impact over Z-pole electroweak precision observables (EWPOs) if Z ′ 23 lives at the TeV scale.We find bounds coming from tests of Z boson lepton universality and flavour-violating Z decays not to be competitive with the bound from ρ.The electron asymmetry parameter A e , which already deviates from the SM by almost 2σ [39], is expected to deviate further in our model.Nevertheless, we expect our model to improve the fit of A FB b , which is in tension with the SM prediction by more than 2σ [39].In conclusion, the global effect of our model over EWPOs can only by captured by performing a global fit, which we leave for future work.In this direction, a dedicated phenomenological analysis along the lines of [21,40] would be much needed.Global fits of EWPOs in the context of other Z ′ models have been performed in the literature, see e.g.[41][42][43], which obtain 95% CL maximum values of sinθ Z−Z ′ ranging from 0.002 to 0.0006 depending on the model.We expect our model to lie on the more restrictive side of that range.In principle, our model could explain the anomalous CDF M W measurement, however in this case we expect the contributions to other EWPOs to be intolerably large, worsening the global fit.

The massive Z ′
23 boson has sizable couplings to light quarks and light charged leptons unless g 3 is very large, which we do not expect based on naturalness arguments and also to protect the extrapolation of the model in the UV, as discussed in Section 5.1.Consequently, on general grounds we expect a significant production of a TeV-scale Z ′ 23 at the LHC, plus a sizable branching fraction to electrons and muons.We have prepared the UFO model of Z ′ 23 using FeynRules [44], and then we have computed the Z ′ 23 production cross section for 13  with the default PDF NNPDF23LO.We estimated analytically the branching fraction to electrons and muons, and we computed the total decay width via the narrow width approximation.We confront our results with the limits from the most recent dilepton resonance searches by ATLAS [46] and CMS [47] in order to obtain 95% CL exclusion bounds.The bounds from ditau [48] and ditop [49] searches turn out not to be competitive even for the region of large g 3 , where the bound from the ρ electroweak parameter is stronger.Our results are depicted as the black-shaded region in Fig. 3.As expected, the bounds become weaker in the region g 3 > 1 where the couplings to light fermions become mildly suppressed.In the region of small g 3 we find the opposite behavior, such that LHC limits can exclude Z ′ 23 as heavy as 6-7 TeV.After combining the LHC exclusion with the bounds coming from the ρ electroweak parameter, we conclude that we can find Z ′ 23 as light as 3.5 TeV for g 3 = 1, while for the benchmark g i ≃ √ 3g Y we obtain M Z ′ 23 5 TeV.
Given that Z ′ 23 has sizable couplings to electrons, we have studied the bounds over contact interactions obtained at LEP [50].For our model, the most competitive bounds arise from contact interactions involving only electrons.Assuming vector-like interactions, the bounds by LEP are only sensitive to regions with very small g 3 , but can exclude Z ′ 23 masses beyond 10 TeV.However, we expect this bound to be slightly overestimated for our model, since the interactions of Z ′ 23 are not exactly vector-like due to the different hypercharge of e L and e R , as depicted in Eq. (5.53).Nevertheless, the bounds over chiral operators are much weaker than the bound over the vector-like operator, and a dedicated reanalysis of the data would be required in order to obtain the proper bound for our model, which is beyond the scope of this work.Therefore, we prefer to be conservative and depict the largest bound of the vector-like operator as the purple region in Fig. 3.
We have also considered implications for B-physics.The heavy boson Z ′ 23 has a sizable lefthanded b L s L coupling and an approximately vector-like and universal coupling to electron and muon pairs.Given these features, a Z ′ 23 with a mass of 2 TeV mediates a meaningful contribution to the effective operator O ℓℓ 9 = (s L γ µ b L )( lγ µ ℓ) (with ℓ = e, µ), where sizable NP contributions are preferred according to the most recent global fits [38], without contributing to the SM-like R K ( * ) ratios [51].However, as depicted in Fig. 3, the region where the model could address the anomalies in b → sµµ data are in tension with the bounds obtained by dilepton searches, as expected for a Z ′ which has sizable couplings to light quarks.Nevertheless, we can see that a relevant C ℓℓ 9 ∼ 0.1 can be obtained for a heavier Z ′ 23 in the region where g 3 < 0.5, as in this region couplings to light fermions are enhanced.B(B s → µµ) is also enhanced above the SM prediction due to both Z ′ 23 and Z exchange diagrams.In the region of small g 3 the Z ′ 23 exchange dominates, while for large g 3 the Z exchange dominates.As anticipated before, the couplings of the Z ′ 23 boson to muons are approximately, but not completely, vector-like.Therefore, a small contribution to the operator O µµ 10 = (s L γ µ b L )(μγ µ γ 5 µ) is generated in the region of small g 3 , where Z ′ 23 couplings to muons are larger, leading to Z ′ 23 exchange being dominant.However, in the region of large g 3 , the flavour-violating coupling sL b L Z ′ 23 mixes into sL b L Z via Z − Z ′ 23 mixing.This provides an effective contribution to O µµ 10 mediated by the Z boson that enhances B(B s → µµ) above the SM prediction.Overall, the region excluded at 95% CL by the current HFLAV average [52] (including the latest measurement by CMS [53]) is depicted as blue-shaded in Fig. 3.It is clear that the resulting bound is not currently competitive with that from the ρ electroweak parameter which constrains the size of Z − Z ′ 23 mixing.The flavour-violating structure of Z ′ 23 fermion couplings leads to sizable contributions to B s − Bs meson mixing [54] and CLFV processes involving τ → µ transitions and τ → e transitions, although well below existing experimental limits [1].τ → µ(e) transitions arise from mixing angles connected to the flavour hierarchy O(m 2 /m 3 ) (O(m 1 /m 3 )), see Section 3.1.As an example, in Fig. 3 we depict the contour for B(τ → 3µ) = 10 −12 .We find τ → µγ to be more competitive than τ → 3µ only in the region g 3 > 1.
Beyond indirect detection, in the near future Z ′ 23 could be directly produced at the LHC, HL-LHC and future colliders such as FCC or a high energy muon collider.The particular pattern of 23 fermion couplings will allow to disentangle our model from all other proposals.For the natural benchmark g 1 ≃ g 2 ≃ g 3 ≃ √ 3g Y , Z ′ 23 preferentially decays to top pairs and ditaus, as can be seen in Table 2. Furthermore, Z ′ 23 preferentially couples and decays to right-handed charged fermions, given their larger hypercharge with respect to left-handed charged fermions.Similarly, decays to downtype quarks are generally suppressed with respect to (right-handed) up-type quarks and charged leptons, given the smaller hypercharge of the former.Due to the modification of EWPOs, our model can also be tested in an electroweak precision machine such as FCC-ee.An alternative way of discovery would be the detection of the hyperon scalars breaking the U (1) 3  Y group down to SM hypercharge, however we leave a study about the related phenomenology for future work.In the same spirit, the model naturally predicts SM singlet neutrinos which could be as light as the TeV scale, with phenomenological implications yet to be explored in a future work.

Conclusions
We have proposed a tri-hypercharge (TH) embedding of the Standard Model, based on assigning a separate gauged weak hypercharge to each family.The idea is that each fermion family i only carries hypercharge under a corresponding U (1) Y i factor.This ensures that each family transforms differently under the TH gauge group U (1) 3  Y , which avoids the family repetition of the SM and provides the starting point for a theory of flavour.
The three family specific hypercharge groups are spontaneously broken in a cascade symmetry breaking down to SM hypercharge.We have motivated a particular symmetry breaking pattern, where in a first step U (1) Y 1 × U (1) Y 2 is broken down to its diagonal subgroup at a high scale v 12 .The remaining group U (1) Y 1 +Y 2 × U (1) Y 3 is broken down to SM hypercharge at a scale v 23 .This symmetry breaking pattern sequentially recovers the accidental flavour symmetry of the SM, providing protection versus FCNCs that allows the NP scales to be relatively low.Dynamics connected to the scale v 12 play a role in the origin of the family hierarchy m 1 /m 2 , while dynamics connected to the scale v 23 play a role in the origin of m 2 /m 3 .The hierarchy of scales v 23 /v 12 generally plays a role on the origin of flavour hierarchies as well, although we have found that a mild hierarchy v 23 /v 12 ≃ λ is enough for specific implementations of the model, where λ ≃ 0.224 is the Wolfenstein parameter.
Assuming that the SM Higgs only carries third family hypercharge, then only the third family Yukawa couplings are allowed at renormalisable level.This explains the heaviness of the third family, the smallness of V cb and V ub quark mixing, and delivers an accidental U (2) 5 flavour symmetry in the Yukawa sector acting on the light families, which provides a reasonable first order description of the SM spectrum.However, U (2) 5 does not explain the hierarchical heaviness of the top quark with respect to the bottom and tau fermions.Furthermore, we have proven that the model generates a similar mass hierarchy for all charged sectors, being unable to explain the heaviness of the charm quark with respect to the strange and muon without small couplings.We have motivated the addition of a second Higgs doublet as a natural and elegant solution, which allows a more natural description of the hierarchies between the different charged fermion sectors.
We have explored the capabilities of the U (1) 3  Y model to explain the observed hierarchies in the charged fermion sector, via the addition of non-renormalisable operators containing U (1) 3  Ybreaking scalars which act as small breaking effects of U (2) 5 .After extracting model-independent considerations from the spurion formalism, we have presented example models where all charged fermion masses and mixings are addressed.Following a similar methodology, we have studied the origin of neutrino masses and mixing in the TH model.We have shown that due to the U (1) 3  Y gauge symmetry, the implementation of a type I seesaw mechanism naturally leads to a low scale seesaw, where the SM singlet neutrinos in the model may be as light as the TeV scale.We have provided an example model compatible with the observed pattern of neutrino mixing.
Finally, we have performed a preliminary exploration of the phenomenological implications and discovery prospects of the U (1) 3  Y theory of flavour.The heavy gauge boson Z ′ 12 arising from the 12breaking displays completely flavour non-universal couplings to fermions, and generally contributes to ∆F = 2 and CLFV processes.The size of the most dangerous contributions are however modeldependent.In selected specific models provided in this manuscript, we have found that the most dangerous contributions to K 0 − K0 mixing and µ → eγ are strongly suppressed, allowing for Z ′ 12 to be as light as 10-50 TeV.Therefore, the lightest gauge boson Z ′ 23 arising from the 23-breaking can live at the TeV scale, within the reach of LHC and future colliders, since Z ′ 23 avoids bounds from K 0 − K0 mixing and µ → eγ thanks to an accidental GIM mechanism for light fermions.
We find the gauge boson Z ′ 23 to have a rich low energy phenomenology: mixing with the SM Z boson leads to implications for the W boson mass and EWPOs, plus we expect contributions to flavour-violating processes involving the third family, such as τ → 3µ(e) and B s − Bs meson mixing.After our preliminary analysis, we find that current data allows Z ′ 23 to be as light as 3-4 TeV in some regions of the parameter space, the strongest bounds coming from dilepton searches at LHC along with the contribution to the ρ electroweak parameter.In the case of discovery, the particular pattern of Z ′ 23 couplings and decays to fermions will allow to disentangle our model from all other proposals.However, most of the phenomenological consequences are yet to be explored in detail: a global fit to EWPOs and flavour observables will allow to properly confront our model versus current data.An alternative way of discovery would be the detection of the Higgs scalars (hyperons) breaking the U (1) 3  Y down to SM hypercharge, however we leave a discussion about the related phenomenology for future work.In the same spirit, the model naturally predicts SM singlet neutrinos which could be as light as the TeV, with phenomenological implications yet to be explored.The tri-hypercharge gauge group may be the first step towards understanding the origin of three fermion masses in Nature, the hierarchical charged fermion masses and CKM mixing, revealing the existence of a flavour non-universal gauge structure encoded in Nature at energies above the electroweak scale.

A General formalism for the seesaw mechanism
In Section 4 we have motivated that in order to explain the observed pattern of neutrino mixing via a type I seesaw mechanism, it is required to add SM singlet neutrinos which carry family hypercharges (but are SM singlets).Therefore, the latter have to be vector-like in order to cancel gauge anomalies.In all generality, such vector-like neutrinos can get contributions to their mass from the unspecified vector-like mass terms and from Majorana masses given by VEVs of hyperons breaking the U (1) 3  Y group.Both the SM singlet neutrinos N and the conjugate partners N (both LH in our convention) can couple to the active neutrinos ν through non-renormalisable operators involving the hyperons.This way, one can write the general neutrino matrix, In order to match Eq. (A.57) to our example seesaw model of Section 4.2, we define ν as a 3component vector containing the weak eigenstates of active neutrinos, while N and N are 2component vectors containing the SM singlet N and conjugate neutrinos N , respectively.Similarly, in the following we define each of the sub-matrices of Eq. (A.57) in terms of the matter content of Section 4.2, ignoring the O(1) dimensionless couplings: ).For simplicity we take them to be equal, although the same conclusions hold as long as they just differ by O(1) factors, as naturally expected.
• For simplicity, we have assumed a similar vector-like mass for both neutrinos in Eq. (A.60), i.e.M N sol ≈ M Natm ≡ M VL .
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. (A.57) and we can safely apply the seesaw formula as Given the structure of m D L and m D R in Eq. (A.58), the products above involving m D L lead to a hierarchical m ν matrix where only the entries in the third row and column are populated and the others are zero.In contrast, the product m D R m T D R provides a matrix m ν where all entries are populated.Therefore, if M VL ≫ v 23 , then the effective neutrino matrix becomes hierarchical, rendering impossible to explain the observed pattern of neutrino mixing and mass splittings with O(1) parameters.Instead, if M VL is of the same order or smaller than v 23 , i.e.M VL v 23 , then the resulting matrix is in any case a matrix where all entries are populated, which has the potential to explain the observed patterns of neutrino mixing.As shown in the main text, Section 4.

B High scale symmetry breaking
Assuming that the 12-breaking scale is far above the electroweak scale, at very high energies we consider only the factors U (1) Y 1 × U (1) Y 2 with renormalisable Lagrangian (neglecting fermion content and any kinetic mixing 10 for simplicity), where for simplicity we assume only one hyperon ϕ 12 (q, −q), which develops a VEV Expanding the kinetic term of ϕ 12 , we obtain mass terms for the gauge bosons as (B.64) The diagonalisation of the matrix above reveals in the basis of mass eigenstates given by (B.66) Therefore, we obtain a massive gauge boson Z ′ 12µ at the scale v 12 , while Y 1 + Y 2 associated to the gauge boson Y 12µ remains unbroken.These results are trivially generalised for the case of more hyperons.The covariant derivative in the new basis is given by 10 Considering kinetic mixing in the Lagrangian of Eq. (B.62)only leads to a redefinition of either the g1 or g2 couplings in the canonical basis (where the kinetic terms are diagonal).
The fermion couplings in Eq. (5.52) are readily extracted by expanding the fermion kinetic terms applying Eq. (B.67).

C Low scale symmetry breaking
The renormalisable Lagrangian of a theory SU (2 2 ) and ϕ 23 (1) (q,−q) reads (neglecting fermion content and kinetic mixing, although we will consider the effect of kinetic mixing at the end of the section), where the covariant derivatives read and we denote the rotation in Eq. (C.74) as V θ 23 , obtaining to the Higgs boson and to ϕ 23 , respectively.In this basis, the covariant derivatives read The mass matrix in this basis can be block-diagonalised by applying the following transformation where the mixing angle is identified with the usual weak mixing angle as and we denote the matrix in Eq. (C.81) as V θ W 11 , obtaining where we have already identified the massless photon.Now we diagonalise the remaining 2 × 2 sub-block in the limit of small r 2 .We obtain  We can see that the SM Z boson carries a small admixture of the X µ boson, which provides a small shift to its mass as where As expected, the SM Z boson mass arises at order r 2 , with a leading correction from Z − Z ′ 23 mixing arising at order r 4 .Instead, the Z ′ 23 boson arises at leading order in the power expansion, with the leading correction from Z − Z ′ 23 mixing arising at order r 2 .Remarkably, the presence of Z − Z ′ 23 mixing always shifts the mass of the Z boson to smaller values with respect to the SM prediction.
The equations obtained match general results in the literature [27,55,56], which consider scenarios where the starting point is a matrix such as Eq.(C.76) with g F = g X .Our equations match those of these papers when g F = g X (and taking into account that we need to perform an extra rotation θ 23 to arrive to Eq. (C.76)).
In the case that a kinetic mixing term sin χF Assuming that the kinetic mixing parameter sin χ is small compared to unity (e.g.absent at treelevel and generated by loop diagrams), then the dominant effect is always the original gauge mixing and kinetic mixing can be neglected.As an example, the hyperon ϕ 23 (1) (q,−q) charged under both U (1) groups generates kinetic mixing at 1-loop as d sin χ/d log µ = −g 12 g 3 q 2 /(16π 2 ), which leads to sin χ(µ) = g 12 g 3 q 2 log(m 2 ϕ 23 /µ 2 )/(16π 2 ).For the natural benchmark g 12 ≈ 3/2g Y and g 3 ≈ √ 3g Y motivated in Section 5.1, along with typical values q = 1/2 and m ϕ 23 = 1 TeV, we obtain sin χ(M Z ) ≃ 0.002.Neglecting the small Z − Z ′ 23 mixing, the fermion couplings of the Z ′ 23 gauge boson given in Eq. (5.53) are obtained by expanding the fermion kinetic terms in the usual way, using the covariant derivative (where T 3 is the third-component SU (2) L isospin, and we do not include the terms associated to charge currents nor QCD interactions) which is an excellent approximation for all practical purposes other than precision Z boson phenomenology.In that case, one has to consider that the couplings of the Z boson to fermions are shifted due to Z − Z ′ 23 mixing as

Figure 1 :
Figure 1: Diagram showing the different energy scales in tri-hypercharge, along with the approximate flavour symmetries that apply at each scale: v 23 denotes the low scale where U (2) 5 is approximately preserved, and the hierarchy m 2 /m 3 is explained.v 12 denotes the higher scale where U (2) 5 is explicitly broken and the hierarchy m 1 /m 2 is explained.

Figure 3 :
Figure 3: Parameter space of the low scale breaking, where M Z ′ 23 is the mass of the heavy Z ′ 23 gauge boson and g 3 is the gauge coupling of the U (1) Y 3 group.The gauge coupling g 12 is fixed in terms of g 3 and g Y via Eq.(5.54), and we consider the non-generic fermion mixing predicted by Model 2 in Section 3.4.Shaded regions in the plot depict 95% CL exclusions over the parameter space, with the exception of the green (light green) region which is preferred by a global fit to b → sµµ data at 2σ (3σ) [38].The dashed line represents the natural benchmark g 1 ≃ g 2 ≃ g 3 ≃ √ 3g Y motivated in the main text.The dashed-dotted line represents the contour where B(τ → 3µ) = 10 −12 .

23 = qv 23 g 2 12 + g 2 3 ,
(C.89) are the masses of the Z boson in the SM and the mass of the Z ′ 23 boson in absence of Z − Z ′ 23 mixing, respectively.All these results can be generalised for the case of more hyperons or more Higgs doublets.

Table 1 :
Charge assignments of the SM fermions under the TH gauge group.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 right-handed (RH) quarks and leptons (so that they become left-handed).In all cases we consider a 2-component convention.

Table 2 :
Main decay modes of Z ′ 23 for the natural benchmark g 1 ≃ g 2 ≃ g 3 ≃ √ 3g Y .We assume that decays into SM singlet neutrinos are kinematically forbidden or suppressed.
60)where I 2×2 is the 2 × 2 identity matrix.In Eqs.(A.59) and (A.60) above, we have considered the following approximations and assumptions: • We have neglected O(1) dimensionless couplings generally present for each non-zero entry of Eq. (A.59).With this consideration, we find M L = M R after the hyperons develop their VEVs.Furthermore, since the two hyperons appearing in M L and M R participate in the 23-breaking step of Eq. (2.8), we have assumed that they both develop a similar VEV ⟨ϕ ℓ23 ⟩ ≈ ⟨ϕ sol ⟩ ≈ O(v 23 2, by further assuming the mild hierarchies v 23 /v 12 ≃ λ and Λ sol /Λ atm ≃ λ the product m D R m T D R above provides a texture for m ν that can reproduce the observed neutrino oscillation data with O(1) parameters.This argument holds as long as m D L is populated by zeros in at least some of the entries involving L 1 and L 2 , like in our example model of Section 4.2.Instead, in the very particular case where both m D L and m D R are similarly populated matrices and of the same order, then the terms proportional to M VL in Eq. (A.61) can provide an effective neutrino mass matrix with O(1) coefficients in each entry.In this case, M VL > v 23 is possible.Nevertheless, even in this scenario we expect M VL not to be very large, since the smallness of m D L and m D R (that arise from non-renormalisable operators) may potentially provide most of the suppression for the small neutrino masses.Furthermore, this scenario involves the addition of several extra hyperons with very particular charges, making the model more complicated, so we do not consider it.