A realistic U(2) model of flavor

We propose a simple U(2) model of flavor compatible with an SU(5) GUT structure. All hierarchies in fermion masses and mixings arise from powers of two small parameters that control the U(2) breaking. In contrast to previous U(2) models this setup can be realized without supersymmetry and provides an excellent fit to all SM flavor observables including neutrinos. We also consider a variant of this model based on a D6 × U(1)F flavor symmetry, which closely resembles the U(2) structure, but allows for Majorana neutrino masses from the Weinberg operator. Remarkably, in this case one naturally obtains large mixing angles in the lepton sector from small mixing angles in the quark sector. The model also offers a natural option for addressing the Strong CP Problem and Dark Matter by identifying the Goldstone boson of the U(1)F factor as the QCD axion.


Introduction
One of the prominent problems of the Standard Model (SM) is the presence of large hierarchies in fermion masses and mixings.Even neglecting neutrino masses, which might have a different origin, the Yukawa couplings span a range from 10 −6 for the electron up to unity for the top quark.Mixing angles in the quark sector are small and hierarchical, while all mixing angles in the lepton sector are sizable.Explaining these hierarchies is referred to as the "SM Flavor Puzzle" (see e.g.Ref. [1] for a review).
A popular framework to address this problem is in terms of approximate flavor (or horizontal) symmetries.The SM fermions are charged under this symmetry, so that most of the Yukawa couplings are forbidden in the symmetry limit.The flavor symmetry is spontaneously broken by vacuum expectation values of scalar fields (the so-called flavons), which can be used to estimate the Yukawa couplings with a spurion analysis.Within an effective field theory approach, appropriate powers of flavon insertions are needed to make a given Yukawa operator invariant under the flavor symmetry, suppressed by some large UV cutoff scale.The flavon vevs are assumed to be slightly below this cutoff scale, so that SM Yukawas arise from powers of these small order parameters.The effective operators have coefficients that are not predicted by the model, but should not be too large or small, in order to explain all hierarchies with the approximate flavor symmetry alone.
While a plethora of this kind of models have been constructed (see e.g.Ref. [1] and references therein), a particular simple and interesting class of models is based on a U (2) flavor symmetry [2,3].In the original model the flavor quantum numbers are compatible with an SO (10) GUT structure, and therefore viable only in a supersymmetric (SUSY) context (or more generally in models with at least one additional Higgs field, needed to account for the m b /m t hierarchy).Holomorphy together with the U (2) breaking pattern by two spurions then leads to three texture zeros in the quark mass matrices, which imply certain relations between CKM mixing angles and quark masses, in particular V ub /V cb = m u /m c .Unfortunately, this prediction is incompatible with the current experimental precision of V ub and V cb , and this simple and economic model was ruled out [4] with the advent of B-factories.
Therefore modifications of the original model have been proposed in order to modify the model predictions and comply with experimental data.In Ref. [5] a SUSY SO (10) model with a D 3 × U (1) flavor symmetry was studied, which mimicked the original U (2) structure with the three texture zeros, but is also in conflict with present values of CKM elements.A more recent study has been performed in Ref. [6], which has shown that the problematic relation can be fixed by taking flavor quantum numbers compatible only with an SU (5) GUT structure.This allows the presence of large rotations in the right-handed (RH) down sector that correct the predictions, as suggested in Ref. [4].Relaxing the SO (10) structure admits to consider also non-supersymmetric models, and in Ref. [7] such a model was constructed with a charged lepton sector designed to address the (still existing) anomalies in semileptonic B-meson decays.This requires to give up also the SU (5) compatibility, but the model can successfully explain the observed deviations in R K [8] and R K * [9] by the tree-level exchange of a Z boson in the TeV range.In contrast to many Z models that address the anomalies, the couplings to fermions are related to the flavor sector and thus essentially predicted in terms of fermion masses and mixings.
In this work we build upon the previous studies in Refs.[2,6,7] and propose a simple, non-supersymmetric U (2) model of flavor that is compatible with an SU (5) GUT structure.The problematic relations between CKM mixing angles and quark masses are modified due to large mixing angles in the RH down sector, allowing for an excellent fit to CKM angles and quark and charged lepton masses.All hierarchies arise from powers of two small parameters (roughly of the same order) describing the U (2) breaking pattern.We also include the neutrino sector, which in this framework can be straightforwardly reproduced by adding three light SM singlets with suitable U (2) quantum numbers and Dirac masses.The fit to the full SM fermion sector is excellent, and predicts the overall mass scale in the neutrino sector below current cosmological bounds.We further discuss a variant of the U (2) model where the SU (2) factor is replaced by the discrete group D 6 .The breaking pattern and the resulting Yukawa matrices closely resemble the SU (2) case.The only difference is a flipped sign in the 1-2 entry of the mass matrices, which has no effect in the quark and charged lepton sector, but allows to obtain Majorana neutrinos masses from the Weinberg operator.In contrast to the Dirac case the parametric flavor suppression of the neutrino mass matrix is fixed purely by charged lepton charges.Remarkably, this matrix is automatically anarchical, and therefore allows for an excellent fit to neutrino data, again predicting the overall neutrino mass scale in about the same range as in the Dirac case.
Finally we discuss the fate of the U (1) ⊂ U (2) Goldstone boson, which naturally plays the role of the QCD axion and has (flavor-violating) couplings to fermions that are predicted by the flavor model, in the spirit of Refs.[10][11][12].In contrast to single U (1) flavor models, here the additional SU (2) flavor symmetry protects flavor-violating couplings to light generations (much as in SUSY U(2) models [6,7]), so that the resulting axion is mainly constrained by astrophysics and not by precision flavor observables.It is well-known that the axion can be an excellent Dark Matter (DM) candidate for large ranges of the U (1) breaking scale, which here is directly connected to the UV cutoff of the flavor model.In this way the model offers a natural solution for the strong CP problem and the origin of DM.
This paper is organized as follows.In Section 2 we define the U (2) flavor model and discuss the structure of the quark and charged lepton sector before addressing the (Dirac) neutrino sector.We then consider a D 6 × U (1) model in Section 3, which closely follows the U (2) structure and allows to obtain Majorana neutrino masses from the Weinberg operator.In Section 4 we address the Strong CP Problem and Dark Matter within this framework, interpreting the Goldstone boson of the U (1) factor as the QCD axion.We finally conclude in Section 5.In three Appendices we provide more details on the group theoretical structure of D 3 and D 6 D 3 × U (1), include more details about the numerical fit, and discuss an explicit example of the scalar potential generating the flavon vevs in the D 6 × U (1) model.

A Realistic U (2) Model of Flavor
In this section we define our framework and show how hierarchies in the quark and charged lepton sector arise from the U (2) flavor symmetry.After discussing the analytical relations between CKM elements and quarks masses, we perform a numerical fit to masses and mixings.We then address the neutrino sector in the context of Dirac neutrinos and include it in the numerical fit.We conclude this section with a general discussion of the flavor structure of neutrino masses, motivating the D 6 × U (1) flavor model in the next section.

Quark and Charged Lepton Sector
We consider an extension of the SM with a global flavor symmetry U (2) F ≡ SU (2) F × U (1) F under which SM fermions are charged.This symmetry group is assumed to be broken slightly below a UV scale Λ, which sets the relevant mass scale for additional dynamics.We assume that the scale Λ is large enough to safely neglect the impact of these new degrees of freedom on phenomenology.Thus we simply work with an effective theory with cut-off scale Λ that only involves SM fields and spurions that parametrize the breaking of SU (2) F × U (1) F .
The SM fermions have U (2) F quantum numbers that are compatible with an SU (5) GUT structure, i.e. they are specified by the quantum number of the two SU (5) representations 10 = Q, U, E and 5 = L, E. The first two generations transform as a doublet under SU (2) F , the third generation is an SU (2) F singlet and the Higgs field is a singlet under both SU (2) F and U (1) F .Thus, the U (1) F quantum numbers of the SM fermions are specified by four charges {X 10a , X 5a , X 10 3 , X 5 3 } for {10 a , 5 a , 10 3 , 5 3 } with a = 1, 2. It turns out that a successful fit to the observed fermion masses and mixings can be achieved for the following simple choice for U (1) F charges: The breaking of the flavor symmetry is described by two scalar spurions φ and χ, which transform under U (2) F as φ = 2 −1 and χ= 1 −1 .These fields acquire the following vacuum expectation values (VEVs): where we will take ε φ ∼ ε χ ∼ O(0.01).In Table 1 we summarize the field content and the transformation properties under the flavor group.As the fermions are charged under U (2) F , Yukawa couplings require additional spurion insertions in order to be U (2) F -invariant.This leads to non-renormalizable interactions suppressed by appropriate powers of Λ.For example, the resulting Lagrangian in the up-sector, at leading order in ε φ,χ , is given by and similar in the down and charged lepton sector.After inserting the spurion VEVs the cutoff dependence drops out, and Yukawa hierarchies arise from powers of the small parameters ε φ,χ .In this way we get for the up-, down-and charged lepton Yukawa matrices (defined as ) where λ f ij are (in general complex) O( 1) coefficients and we have kept only the leading contributions in ε φ,χ .Note that, in contrast to the supersymmetric U (2) model in Ref. [6], there are no holomorphy constraints, which leads to a more general Yukawa pattern.
One can show that the λ 11 , λ 13 , λ 31 entries give only subleading corrections to quark masses and mixings, which are relatively suppressed by at least ε 2 φ .Thus, effectively, three texture zeros appear in the Yukawa matrix, much as in the supersymmetric models [6], and to good approximation we obtain the Yukawa couplings Because of the hierarchical structure and the presence of the texture zeros, it is possible to analytically derive some approximate results for the singular values and the rotations to the mass basis [7].One can also perturbatively diagonalize the Yukawa matrices, and obtain the following estimates for singular values and CKM matrix elements (neglecting O(1) coefficients): These expressions can be compared to the (1σ) ranges for fermion mass ratios and CKM elements, taken for definiteness at 10 TeV where λ = 0.2 ≈ V us and y b (10 TeV) ≈ λ 2.7 , y τ (10 TeV) ≈ λ 2.8 .Within roughly a factor λ, all hierarchies can be reproduced taking and therefore a good fit to masses and mixings can be expected with input parameters λ f ij that are indeed O(1).Moreover, it is clear that there must be four relations in each fermion sector between the 3 singular values and the 3+3 rotation angles.For real h f ij it is straightforward to work out these predictions exactly [7] and expand the result in ratios of the hierarchical eigenvalues.One can then relate the 1-2 and 1-3 rotations in the left-and right-handed sectors to the 2-3 rotations and the eigenvalues.With the convention where V ij are orthogonal rotation matrices in the i-j plane that are parametrized by the angles s ij ≡ sin θ ij , one obtains up to percent corrections where we included also relative phases φ 1,2 , see Ref. [6] for details.In the original U (2) models in Ref. [2,3], the rotation angle in 2-3 RH down sector s Rd 23 was taken to be of the order of the other 2-3 rotation angles, s Rd 23 ∼ V cb .From the above equations, this directly leads to the accurate prediction |V ub /V cb | ≈ m u /m c which deviates from experimental data by more than 3σ.This is the reason why here this angle is taken to be large, s Rd 23 ∼ c Rd 23 ∼ 1/ √ 2, which then allows to obtain an excellent fit to CKM angles as we demonstrate in the next section (see also Refs.[4,6,7]).

Fit to Quark and Charged Lepton Sector
We now perform a numerical fit to the model parameter set {λ u,d,e ij , ε φ , ε χ }.For simplicity we restrict to real λ u,d,e ij and demonstrate later on that the CKM phase can be obtained by taking a complex parameter λ u 33 .The experimental input parameters are therefore the quark and charged lepton masses and the CKM mixing angles.For concreteness we take them in the MS scheme at 10 TeV from Ref. [13], with a symmetrized 1σ error taken to be the larger one.All input parameters are summarized in  TeV taken from Ref. [13].
The quality of the fit with a given model parameter set {λ u,d,e ij , ε φ , ε χ } is measured by two functions χ 2 and χ 2 O(1) .The first quantity is the usual χ 2 that indicates how well the experimental input values are reproduced by the fit.It is obtained by plugging the model parameters into the Yukawa matrices in Eq. (2.1) and calculating numerically the singular values y q,l and the CKM mixing angles θ ij in the PDG parametrization.These values are used with the experimental input above to obtain χ 2 defined as q=u,d,s,c,b,t (y q − y q,exp ) 2  (σy q,exp ) 2 + =e,µ,τ 12),( 13),( 23) In order to explain Yukawa hierarchies solely by U (2) F breaking, the parameters λ u,d,e ij should be O (1).The meaning of this requirement is somewhat fuzzy, and here we choose to quantify it by introducing a measure χ 2 O(1) defined as where i, j = 1, 2, 3 and p = u, d, e.This corresponds to the assumption that the λ u,d,e ij are distributed according to a log-normal distribution with mean 1 and standard deviation σ = 0.55, i.e. the absolute values λ u,d,e ij lie with a probability of 95 % within the interval [1/3, 3].For example, the contribution to χ 2 O(1) of a single parameter λ = {3, 5, 7, 10, 50, 100} (or the inverse) is ∆χ 2 O(1) = {2, 4, 6, 9, 25, 35}.We consider a fit satisfactory as long as χ 2 O(1) ≤ #pars, and there are 5 parameters for each fermion sector.As the best fit we choose the one that minimizes both χ 2 and χ 2 O(1) .In Table 3 we show our fit results, where we display the values of the small parameters ε φ , ε χ and indicate separately the two fit measures χ 2 , χ 2 O(1) as defined above, along with the smallest and largest |λ u,d,e ij |.For the fit QL1 R we have minimized χ 2 + χ 2 O(1) , while for QL2 R we have minimized χ 2 O(1) while keeping χ 2 ≤ #obs = 12.For illustrative purposes we also show a fit that minimizes just χ 2 (QL3 R ).Indeed there are enough free parameters to obtain a perfect fit to observables, however one needs χ 2 O(1) as large as 35 and O(1) parameters as small as ≈ 1/9, so this fit should be discarded according to our quality requirement χ 2 O(1) < 15.The best fits are QL1 R and QL2 R with O(1) parameters between 1/3 and 3, which feature values of ε φ , ε χ that are indeed of the naive size estimated in Eq. (2.8).
Finally we demonstrate that the CKM phase δ CP can be easily included.For simplicity we restrict to the case where only the 33 entry in the up-quark Yukawa matrix is complex, i.e. λ u 33 → λ u 33 e iδ 33 .In a realistic setup where all Yukawas have phases, the fit can only get better.In the χ 2 measure in Eq. (2.13) we now include the CP phase of the CKM matrix, with the experimental value taken from Ref. [13] δ CP,exp = 1.208 ± 0.054 .
Including δ 33 leads to even better fits (QL1 and QL2), which we show in Table 4.This demonstrates that an excellent fit for quark and charged lepton sector, including the CKM phase, can be obtained with all O(1) parameters lying between 1/2.8 and 2.1.

Neutrino Sector
In the neutrino sector we have to distinguish whether neutrinos are Dirac or Majorana.We begin with the discussion of the Dirac scenario, since the Majorana case in the U (2) F model is strongly disfavored as we will discuss below.To this extent we introduce SM singlets N a , N 3 with U (1) F charges X N a and X N 3 , where N a transforms as a doublet of SU (2) F and N 3 as a singlet.The Lagrangian then allows for a Yukawa coupling L ν = L T Y ν N H (we assume that the Majorana mass term is forbidden, e.g. by exact lepton number conservation).As in the charged lepton sector, one can obtain its structure from a spurion analysis as It is clear that in order to obtain sub-eV neutrinos one needs large U (1) F charges X N a,3 > 1, so that one can drop the absolute values in Eq. (2.15).In this case the contributions from the (11), ( 13), (31) entries to masses and mixings are again sub-leading, and we can drop them as in the previous section and are left with the Dirac neutrino mass matrix (2.16) It is well-known that an anarchical neutrino mass matrix can give a good fit to neutrino observables, which can be achieved taking (2.17) In order to obtain a overall neutrino mass scale 0.1 eV, one needs X N 3 5, so that tiny neutrino masses arise from somewhat large U (1) F charges and the smallness of the U (2) F breaking parameters, ε χ,φ ∼ 0.01.
These considerations are confirmed by a numerical fit, for which we proceed as in the previous section, now including the neutrino sector.For the input values for normal (NO) and inverted mass ordering (IO), we use the neutrino mass differences and PMNS mixing angles from the global NuFIT 3.2 (2018) in Refs.[14,15], which are summarized in Table 5.We then plug the neutrino model parameters λ ν ij for fixed charges X N a , X N 3 into the Yukawa matrices in Eq. (2.17) and calculate numerically the singular values and the PMNS mixing angles θ ij in the standard parametrization.To the χ2 defined in Eq. (2.13) we add the corresponding expression χ 2 ν in the neutrino sector1 12),( 13),( 23)

Normal Ordering (NO)
and similarly we include the coefficients λ ν ij in the measure χ 2 O(1) defined in Eq. (2.14).We then perform a simultaneous fit to quark, charged lepton and neutrino sector including a phase in λ u 33 as discussed in the last section (for simplicity we omit phases in the neutrino sector, including them would make the fit only better).The fit results are shown in Table 6, both for NO and IO.

Fit
X N a QLν As expected, good fits are obtained only for equal charges X N a = X N 3 = 5 ÷ 6.There is clearly a strong preference for NO, as can be seen in both quality parameters χ 2 and χ 2 O(1) (and the smallest/largest λ ij ).According to our quality requirement χ 2 O(1) < 20, we should actually discard the IO possibility, since all fits with inverted mass ordering violate this criterion, and we include it just for illustrative purposes.
Comparing to the fit of quark and charged lepton sector only (cf.Table 4), one can see that including neutrinos make the fits slightly worse, but still with O(1) coefficients between 1/3 and 3.The fits determine all neutrino parameters, and we obtain predictions for the absolute mass scales and two important observables, the sum of masses Σm i as probed by satellite telescopes, and the effective neutrino mass m β = i m 2 i |U ei | 2 as measured in the β-decay spectrum close to the endpoint.All predictions are summarized in Table 7. Table 7: Predictions for neutrino masses and observables for the NO fits in Table 6.
Since in contrast to the quark sector there are predictions for observables that are not yet measured, we also give a range for these predictions scanning over many fits with X N a = X N 3 = 5, 6 on which we only impose the (somewhat arbitrary) condition that χ 2 < 20 and the quality requirement χ 2 O(1) < 20 (which excludes IO).In this way we obtain predictions for the ranges of m i and m β as shown in Table 8, where we also indicate the value preferred in most fits.

Quantity Range [meV] Preferred values [meV]
m i 58 -110 60 -65 m β 8 -26 9 -10 We notice that the predicted range for m β is an order of magnitude below the expected future sensitivity of m β 0.2 eV by the KATRIN experiment [16].The prediction for the neutrino mass sum is consistent with present bound by PLANCK giving m i < 0.23 eV [17,18] and in the reach of the EUCLID satellite that is expected to measure m i with an error of about 0.05 eV [19,20].Note that the lower bound on the predicted range of m i essentially saturates the minimal value that is obtained for a massless lightest neutrino, which (including 1σ errors) is given by 58 meV for normal ordering.Finally, we discuss the case of Majorana Neutrinos.In addition to the neutrino Yukawa coupling, the Lagrangian contains a Majorana mass term, The Yukawa matrix Y ν is the same as in Eq. (2.15), while the Majorana mass matrix can be obtained as where we factored out a single mass scale M that is taken of the order of the usual see-saw scale, M ∼ 10 14 GeV.One can therefore integrate out the heavy singlets and get light neutrino masses from the Weinberg operator y ij /M (L i H)(L j H), according to the type-I seesaw formula Notice that the 1-2 entry of M ν without any φ insertion vanishes because of the necessary SU (2) anti-symmetrization, and therefore picks up an additional ε 2 φ suppression.It turns out that this extra suppression spoils the naive EFT spurion analysis of the Weinberg operator using only the charges of L a , L 3 (since negative powers of φ appear in the UV theory), and one has to use Eq.(2.20) to calculate m M ν .We first assume that X N A ≥ 1 and X N 3 ≥ 0, so that one can drop the absolute values and obtain for the parametric structure of the light neutrino mass matrix where ε ∼ ε φ ∼ ε χ (notice that the charges X N a,3 drop out).Such a structure is clearly ruled out, since it gives singular values {ε 2 , ε 2 , 1/ε 2 }, which would imply normal hierarchy along with a parametric prediction for the ratio of mass differences ∆m 2  21 /∆m 2 31 ∼ ε 4 × ε 4 that is way too small.Moreover, one can check that also different charge assignments for X N a,3 do not allow to obtain a Majorana neutrino mass matrix that leads to a good fit, besides losing predictivity.Indeed the main theoretical advantage of Majorana neutrinos over Dirac neutrinos would be a scenario in which the effective Majorana mass matrix does not depend on the details of the UV physics, i.e. the choice of X N a,3 .
We conclude this section with the observation that the Majorana scenario would work perfectly if not for the vanishing of the leading 1-2 entry in the heavy mass matrix in Eq. (2.19).Indeed, if this entry would be given by κ 12 ε , and X N a ≥ 1, X N 3 ≥ 0, the effective light neutrino mass matrix would be given by (the dependence on X N a,3 drops out again) which apart from the subleading 11, 13, 31 entries has only very mild ε χ /ε φ hierarchies and suggests a very good fit to neutrino observables.Note this absence of hierarchies is actually a prediction of the quark and charged lepton sector, which requires equal charges for the lefthanded doublets L a and L 3 , and order parameters of similar size, ε χ ∼ ε φ .If therefore the 1-2 elements were symmetric instead of anti-symmetric, all low-energy mass matrices would follow the same hierarchical pattern, differing only in the U (1) F charge assignment of the third generation, which is 0 for Q 3 , U 3 , E 3 and 1 for D 3 , L 3 .Thus the light 2 × 2 sub-block would be the same in all fermion sectors, and only the third coloum/row would differ by powers of ε χ , giving where we neglected the mild ε χ /ε φ hierarchy that is responsible for e.g. the Cabibbo angle.As we discuss in the following section, this simple pattern allows for an excellent fit to all fermion observables, and the necessary 1-2 symmetric structure can be obtained when considering the (discrete) dihedral group D 6 instead of SU (2) as flavor symmetry, which closely resembles the SU (2) structure apart from a sign flip in the 1-2 entries.

A D 6 × U (1) Model of Flavor
In this section we consider the same framework with a D 6 × U (1) flavor symmetry, which closely resembles the U (2) case.We first introduce some D 6 D 3 × Z 2 group theory and discuss the resulting flavor structure of quark, charged lepton masses and the Weinberg operator.
After some brief analytical considerations for the resulting predictions for neutrino observables, we perform a numerical fit to all fermion observables and conclude with a discussion of the phenomenological implications.

Setup
As we have just discussed, we want to mimic the structure of U (2) within a discrete flavor group that allows for a symmetric singlet contraction of two doublets.The simplest such group is the dihedral group D 3 , the symmetry group of an equilateral triangle, which is discussed in detail in Appendix A. This group is actually a subgroup of SO(3) and not of its double cover SU (2), and it is isomorphic to the permutation group S 3 .It features two one-dimensional representations 1 and 1 and one two-dimensional representation 2. The contraction of two doublets ψ = (ψ 1 , ψ 2 ) and φ = (φ 1 , φ 2 ) into the singlet 1 is given by Therefore we could simply assign the SM and spurion fields to D 3 representations that follow the SU (2) ones, i.e. the doublets 10 a , 5 a , φ a are in a 2 of D 3 and all other fields are total singlets.However, in contrast to SU (2) the product of two doublets also containts a doublet, so that three doublets can be contracted to a singlet as This implies that in contrast to the SU (2) model a large 1-1 entry is generated, for example in the up-sector by the operator which would be no longer negligible and thus would completely spoil the hierachical structure.
In order to suppress this entry, we would like to mimic the SU (2) structure in which such a contraction is forbidden by the Z 2 center of SU (2), under which the doublets are odd and the singlet is even.Therefore we consider2 D 3 ×Z 2 which is isomorphic to D 6 , the symmetry group of a regular hexagon (see Appendix A for details), and finally make the charge assignment as in Table 9.The additional Z 2 factor ensures that the contraction of three 2 − doublets does not contain the total singlet 1 + , and in the quark and charged lepton sector we obtain the very same spurion analysis as for U (2) in Section 2 (see Eq. (2.4)), except for the sign in the 1-2 entry: ) In the neutrino sector we work with the effective Weinberg operator y ij /M (L i H)(L j H), which can be induced by the type-I seesaw mechanism as discussed in the previous section.Its parametric structure is predicted in terms of the D 6 ×U (1) F quantum numbers of the charged leptons, which gives for the light Majorana neutrino mass matrix Here we have used the same vacuum expectation values as before although in contrast to the SU (2) F case we cannot use D 6 transformations in order to assume this VEV for φ without loss of generality.Therefore we provide an explicit scalar potential in Appendix C with only one additional scalar field that generates dynamically the above VEVs3 .Altogether we obtain to good approximation the mass matrices ) As discussed in the previous section, this model has the remarkable feature that the hierarchies in the quark and charged lepton sector require ε φ ∼ ε χ , and therefore naturally gives rise to an approximately anarchic neutrino mass matrix with generically large mixing angles.Before we perform a numerical fit, we proceed with some analytical considerations.In the quark and charged lepton sector the analysis of the previous section is unaltered, since the flipped sign in the 1-2 entry does not play a role at leading order.In the neutrino sector we have 4 real parameters, which will enter the PMNS matrix together with three charged lepton rotations angles controlled by a single free real parameter s Le 23 , see Eq. (2.10).These parameters correspond to 5 observables (3 PMNS angles + 2 squared mass differences), so up to phases all parameters are fixed and one can predict the absolute neutrino mass scales and related observables.There are 4 phases in the neutrino sector and 2 phases in the lefthanded charged lepton rotations, which combine to 3 physical phases, one Dirac and two Majorana phases.To study the prediction of the overall neutrino mass scale, we parametrize the neutrino mixing matrix V ν (defined by ) in the standard CKM form multiplied with a phase matrix P ν = diag(e iα 1 , e iα 2 , 1) from the right and a phase matrix P from the left.Inverting the defining equation, we get from the vanishing 11 and 13 entries the two equations The angles in the neutrino sector s ij,ν are connected to the observed PMNS mixing angles through V PMNS = (V e L ) T V ν .Since the 1-2 rotation in the charged lepton sector is small, ∼ m e /m µ ≈ 0.07, we have to good approximation s 12,ν ≈ s 12 , but 2-3 rotations in the charged lepton sector are large, so that both θ 23 and θ 13 generically receive large contributions from the charged lepton sector.Nevertheless one can easily verify that Eq. (3.10) cannot be satisfied for inverted mass ordering, while for normal ordering one can obtain an upper bound on the lightest neutrino mass m 1 , by maximizing the neutrino mixing angles s 12,ν and s 13,ν with a suitable choice of phases.If one neglects the charged lepton contribution to s 12 , one can show that m 1 ≤ 11 meV, which in turn leads to upper bounds m i ≤ 76 meV, m β ≤ 14 meV and m ββ ≤ 13 meV.This estimates is confirmed by the numerical analysis in the next section.

Numerical Fit
We now perform a simultaneous fit to quark, charged lepton and neutrino sector including a phase in λ u 33 as in the last section (for simplicity we omit phases in the neutrino sector, including them would make the fit only better).The fit results are shown in Table 10, and include also the effective suppression scale M of Weinberg operator, which is of the order of 10 11 GeV.The fit is even better compared to Dirac Neutrinos (cf.Table 6), with all O(1) parameters roughly between 0.4 and 2. Fit M [10 The complete set of parameters can be found in Table 15.
The corresponding prediction for the neutrino masses m i , its sum m i , the neutrino mass m β and the "effective Majorana mass" m ββ = U 2 ei m i measured in neutrinoless doublebeta decay are shown in Table 11.As expected from the analytical considerations, only a normal hierarchy for the neutrino masses is viable.The predicted values for m i and m β are similar to the ones in the Dirac Neutrino case (cf.Table 7), while the effective Majorana mass is well below the expected sensitivities even in near future neutrinoless double-beta decay experiments [21].
Fit Finally we also give a range for the observables scanning over many fits on which we only impose that χ 2 < 20 and χ 2 O(1) < 20.In this way we obtain predictions for m i , m β and m ββ lying in the ranges shown in Table 12, where we also indicate the value preferred in most fits.This result agrees well with our estimate in the last section, where we have also included phases, so we expect the upper bounds on the mass scales to be approximately valid even when including phases in the numerical fit (the lower bounds again saturate the limit obtained from taking the lightest neutrino massless).We conclude this section with a discussion of the phenomenological implications of our model.As we have seen, the flavor sector itself gives rise to quite narrow predictions for observables in the neutrino sector, which are however far below the present experimental sensitivities.In order to obtain other experimental signals, we have to rely on new low-energy dynamics besides the SM.The natural candidate for such new degrees of freedom are the fields at the cut-off scale Λ, which we have not specified so far (in particular the radial components of the flavons φ and χ naturally get a mass at that scale).However, effects of these fields and other dynamics related to the UV completion are suppressed by powers of 1/Λ, and there is no reason that Λ is sufficiently close to the electroweak scale in order to give rise to sizable deviations from the SM.Still it would be interesting to consider an explicit UV completion of the present model to study the structure of these effects in detail.

Quantity Range [meV] Preferred values [meV]
Another option for light dynamics, which is essentially model-independent and wellmotivated, is provided by the pseudo-scalars in the flavon fields.If there is no explicit breaking of the U (2) F symmetry, the associated Goldstone bosons are exactly massless, apart from a linear combination that can be identified with the QCD axion, which solves the strong CP problem and gets a mass from non-perturbative effects.The easiest way to get rid of the orthogonal massless Goldstones is replacing SU (2) F by a discrete subgroup, which is another advantage of the D 6 × U (1) model discussed in this section.In this case there a single Goldstone boson associated with the U (1) F factor that can naturally serve as the QCD axion, as we are going to discuss in the next section.

The U (2) Axiflavon
As originally proposed in Ref. [10], a Goldstone boson arising from the breaking of global flavor symmetries could play the role of the QCD axion.Indeed any Goldstone of a U (1) symmetry with a QCD anomaly will solve the strong CP problem, and one can demonstrate (see Ref. [11]) that there is a non-zero SU (3) c × SU (3) c × U (1) F anomaly in any flavor model where the determinants of up-down and down-quark mass matrices are controlled dominantly by the U (1) F symmetry factor.In the present model this is indeed the case as det m u ∼ ε 4 χ and det m d ∼ ε 5 χ , due to the presence of the approximate texture zeros, see Eq. (3.8).Moreover, if also the determinant of the charged lepton mass matrix depends only on the U (1) F breaking, the ratio of electromagnetic and color anomaly coefficients E/N is expected to be a rational number close to 8/3 [11].In the present model the U (1) F charge assignment is actually compatible with SU (5), so it is clear that we get exactly E/N = 8/3, as in minimal DFSZ [22,23] and KSVZ models [24,25], and thus the same axion couplings to photons.
In this section we will calculate the axion couplings and fermions, concentrating on the flavor-violating couplings to fermions, which follow from the hierarchical structure of fermion masses and mixings.In particular axion couplings to nucleons and electrons are fixed in terms of the U (1) F charges, while flavor-violating couplings to quarks and leptons are controlled by the unitary rotations that diagonalized the Yukawa matrices.Their parametric suppression is determined by the U (2) F quantum numbers, and their numerical value by the fit to fermion masses and mixings.We then study the phenomenology of this axion, finding that the strongest constraints on the axion mass (or equivalently the U (1) F breaking scale) come from astrophysical constraints (as in the minimal DFSZ and KSVZ models), since flavor-violating axion couplings to light quarks are strongly suppressed by the approximate SU (2) F structure.

Axion Couplings
We begin by identifying the axiflavon as the Goldstone boson arising from the spontaneous breaking of U (1) F induced by the vevs of φ and χ.In general the Goldstone is a linear combination of the phases a i of the scalar fields φ i with charge X i and (real) vev V i , given by Thus we find that χ and φ contain the Goldstone as (we ignore the radial mode) where we have defined the U (1) F breaking scale V ≡ ε 2 χ + ε 2 φ Λ.The couplings of a to fermions can be obtained by inserting the above expressions for χ and φ into the effective Yukawa Lagrangian given by Eq (2.3) for the up sector and the analogous terms in the down-and charged lepton sector.It is then convenient to change field basis by performing a U (1) F transformation of the fermion fields which will remove the a(x) dependence from the Yukawa sector, because of U (1) F invariance.Since this transformation is anomalous, it will generate axion couplings to gauge field strengths, and since it is local it will modify fermion kinetic terms.The resulting couplings to gluon and photon fields strengths are given by with the dual field strength Fµν = 1 2 ε µνρσ F ρσ and the anomaly coefficients Thus we obtain E/N = 8/3 exactly, which is just a consequence of the fact that the U (1) F charge assignment is compatible with SU (5).The modification of fermion kinetic terms leads to axion-fermion couplings in the flavor interaction basis In the mass basis, defined as with Finally we switch to Dirac spinor notation for the fermions and introduce f a ≡ V /( √ 2N ) to match to the standard normalization for the anomalous couplings.These are given by with E/N = 8/3 in this model (and domain wall number N DW = 2N = 9).The couplings to fermions are given by with and the shorthand notation Note that the diagonal elements of these parameters satisfy While the above expressions are valid for any axion model with PQ charges that are universal for two fermion generations 4 , in the present model these expressions simplify to Using the approximate expressions in Eq. (2.10), the rotations have the parametric structure so that all relevant V 3i are CKM-like, and we have Therefore the diagonal axial couplings are to very good approxmation independent of the rotations, and we get, denoting The flavor-violating axion couplings are controlled by ε f ij , whose numerical values, beyond the parametric suppression given above, are known for a given fit to masses and mixings.Besides there is an overall suppression factor 1/f a that is proportional to the axion mass m a , with the usual conversion factor for QCD axions [27] m a = 5.7 µeV 10 12 GeV f a .(4.23)

Axion Phenomenology
The most important constraints on fermion couplings of invisible (stable) axions (cf.Eq. 4.12) are summarized as an upper bound on the quantity (m a /coupling) in the first column of Table 13.These include flavor-violating b − s transitions as tested in B → Ka decays [28], flavor-violating s − d transitions contributing to K → πa decays [29], (flavor-diagonal) axionelectron couplings bounded by the measurement of the WD luminosity function [30], and effective axion couplings to nucleons constrained from the burst duration of the SN 1987A neutrino signal [31]. Coupling Table 13: Bounds on selected axion-fermion couplings; here C N ≡ C 2 p + C 2 n denotes the effective couplings to nucleons, with axion couplings to protons and neutrons C p,n defined analogously to the axial vector couplings in Eq. (4.12).The second column denotes the modelindependent upper bounds on the ratio of m a /C, where C denotes the respective coupling, while the third and fourth columns contain the upper (lower) bound on m a (f a ) in our model, using the numerical results for the couplings of Section 3.2, where for explicitness we took the fit QLν M -1 (other fits give similar constraints).
We have further used the predictions of the axion couplings in our model to obtain an upper bound on m a , or equivalently a lower bound on f a , which is shown in Table 13 for the fit QLν M -1 of the complete D 6 × U (1) model in Table 10 (the result for the other fits are very similar).As a result of the strong CKM protection of s − d transitions C V sd ∼ λ5 , the main constraint on the model comes from astrophysics, similar to flavor-universal axion models.Since the bound from WD cooling and SN1987A are comparable, and the precise value of the latter is debated in the literature (see e.g. the recent discussion in Ref. [32] which finds a constraint on m a /C roughly a factor 5 weaker than the PDG bound), we only take the constraint from WD cooling, giving a upper bound on the axion mass m a < 14 meV.This translates into a lower bound on the cutoff Λ > 1.9 • 10 10 GeV.The predictions for the branching ratio of K + → π + a decays are given which is far below the future sensitivity of NA62 [33,34] given the constraint from WD cooling.This is in sharp constrast to the U(1) Axiflavon proposed in Ref. [11] (see also Ref. [12]), where the d − s transition is only Cabibbo-suppressed, C V sd ∼ λ, so that K + → π + a provides the strongest constraint on the axion mass.
The upper bound on m a < 14 meV implies that the axion is stable on cosmological scales.It is a remarkable feature of the QCD axion that it can also explain the observed Dark Matter (DM) abundance.One of the simplest scenarios is the misalignment mechanism [35][36][37], valid when U (1) F is broken before inflation 5 .At this stage the axion is essentially massless and takes a generic field value misaligned from the vacuum value by an angle θ.Around the QCD phase transition the axion potential is generated, and the axion begins to oscillate around the minimum.The energy density stored in these oscillations can be approximately related to the present DM abundance as [17] Ω DM h 2 ≈ 0.12 6 µeV m a 1.165 where θ ∈ [−π, π] is the initial misalignment angle.Thus for not too small values θ 0.1π, the natural window for axion DM is given by axion masses roughly between (1 ÷ 40) µeV, which correspond to a axion decay constant f a ∼ (10 11 ÷ 10 13 ) GeV and a cutoff in the range 6 Λ ∼ (10 13 ÷ 10 15 ) GeV.This range of axion masses preferred by DM through the misalignment mechanism will be probed by the ADMX upgrade in the near future [38].Indeed the discovery prospects of the U (2) Axiflavon are mainly due to its coupling to photons, and we summarize the status of the relevant experiments in the usual (m a , g aγγ ) plane in Fig. 1, where g aγγ = |8/3 − 1.92| α em /(2πf a ).
suitable explicit breaking term to solve the domain wall problem arising from NDW = 9.This is in contrast to the U(1) Axiflavon in Ref. [11], where the upper bound on the axion mass from K → πa prevents to obtain the right amount of axion dark matter if U (1)F is broken after inflation. 6Repeating the numerical fit as in Section 3.2 with SM input values at 10 14 GeV, the χ 2 and χ 2 O (1) get slightly worse (0.4/11 and 18/9.1 compared to 0.7/7.9 and 18/6.3 at 10 TeV, see Table 10), while the overall predictions change only marginally.

Summary and Conclusions
In summary we have a proposed a U (2) F model of flavor with horizontal quantum numbers compatible with an SU (5) GUT structure.The flavor symmetry U (2) F = SU (2) F × U (1) F is spontaneously broken by two flavon fields φ and χ, which transform as a doublet and singlet under SU (2) F , respectively.Similarly, the three generations of SM fermions transform as 2 + 1 of SU (2) F , and there is a simple assignment of U (1) F quantum numbers (5.26) The SM Yukawas arise from higher-dimensional operators made invariant under U (2) F by appropriate insertions of flavons, suppressed by the cut-off scale Λ v.In this way the hierarchical structure of Yukawa matrices arise from powers of two small parameters that control the breaking of U (2) F by the flavon vevs, up to Wilson coefficients that are required to be O(1).The resulting Yukawa matrices in the quark and charged lepton sector have a simple structure with three texture zeros in the 1-1,1-3 and 3-1 entries, while the 1-2 entry is antisymmetric, see Eq. (2.1).The presence of these textures leads to accurate relations between CKM elements and masses, cf.Eq (2.12), which in contrast to the original U (2) flavor models in Refs.[2,3] can be consistent with experimental data because of large rotations in the right-handed down quark sector.Indeed we have obtained a very good fit to fermion masses and mixings with coefficients that are indeed O(1) (all between 0.4 and 2), see Table 4.
We have then included the neutrino sector, which gives a consistent fit to experimental data only with Dirac neutrinos.To this extent we have introduced three right-handed neutrinos (SM singlets), which also transform as 2 + 1 of SU (2) F and have equal charges under U (1) F .The resulting structure of the Dirac mass matrix (cf.Eq. (2.17)) has again three texture zeros and only weak inter-generational hierarchies, thus predicting large mixing angles.The U (1) F charge of the singlets enters only in the overall suppression factor and can account for the smallness of neutrino Yukawas if taken to be 5 ÷ 6.The combined fit to the complete fermion sector is viable only for neutrinos with normal mass hierarchy, and still shows a good performance with O(1) coefficients between roughly 1/3 and 3 (cf.Table 6).This fit determines all parameters in the neutrino sector, and thus gives predictions for the absolute neutrino mass scale and the related observables.Scanning over many good fits we have obtained a range for the sum of neutrino masses roughly given by (58 ÷ 110) meV, while the prediction for the effective neutrino mass measured in β-decays is far below future experimental sensitivities.
In order to have a consistent scenario with Majorana neutrinos, we have futhermore discussed an D 6 × U (1) variant of the U (2) F model, where the SU (2) F factor is replaced by a discrete D 6 subgroup.The charge assignment of fermions and spurions closely resembles the U (2) F structure, so that the effective Yukawa matrices in the quark and charged lepton sector are exactly the same as in the U (2) F case, up to a sign flip in the 1-2 entry that is largely irrelevant.This sign flip however allows for an unsuppressed 1-2 entry in the Weinberg operator, whose hierarchical structure follows directly from charges of the SM lepton doublets, and are to large extent independent of the charges of the heavy right-handed neutrinos (cf.Eq. (2.22)).Remarkably, the resulting structure automatically leads to an anarchic neutrino mass matrix, so that the SU (5) structure connects large leptonic mixing angles to small mixing angles in the quark sector.Indeed the parametric flavor suppressions of up-, down-quark, charged lepton and neutrino masses follow the simple pattern where the mass scale is set by v in the quark and charged lepton sector and v 2 /M in the neutrino sector.The difference between the fermion sectors just follows from the different U (1) charge assignment for the third generation, see Eq. (5.26).Although this model is more predictive than the Dirac case, since two U (1) charges are replaced by a single mass scale M , we obtain an excellent fit to all SM observables with O(1) coefficients between 0.4 and 2, see Table 10.From this fit we can again predict the overall neutrino mass scales, and as in the previous case only neutrinos with normal mass hierarchy are viable.Scanning over many good fits we have obtained a slightly narrower range for the sum of neutrino masses roughly given by (58 ÷ 78) meV, while again the predictions for the effective neutrino mass entering beta decay and neutrinoless double beta decay are far below future experimental sensitivities, see Table 12.
Finally we have discussed the various possibilities to test our models apart from the predictions in the neutrino sector.In general sizable deviations in experimental observables from the SM require the existence of sufficiently light degrees of freedom.While there is no particular reason why the cutoff and its associated dynamics should be light, there is the natural possibility to solve the strong CP problem and account for DM through the Goldstone boson of the global U (1) F symmetry, which we refer to as the U (2) Axiflavon.In contrast to the Axiflavon related to a single Froggatt-Nielsen U (1) F symmetry as presented in Refs.[11], here the flavor-violating couplings of the axion are protected by the approximate U (2) symmetry.Therefore the U (2) Axiflavon looks very much like a usual DFSZ/KSVZ axion, with the strongest constraint from WD cooling, which requires a sufficiently light axion m a < 14 meV.Particularly interesting is the axion mass range where DM can be explained through the misalignment mechanism, implying axion masses around (1 ÷ 40) µeV, which corresponds to a cutoff scale of roughly (10 13 ÷ 10 15 ) GeV.This range will be tested by future axion haloscope searches.
The present model could be extended in several ways: 1) A more careful study of the neutrino sector might allow to pin down the predictions analytically, and it could be interesting to take a closer look to the type-I seesaw model, in particular its connection with Leptogenesis 2) One could embed the model into a supersymmetric framework to address the hierarchy problem 3) Finally it might be interesting to study possible UV completions and calculate the low-energy constraints from flavor-violating obervables on the new dynamics.
For the case of ψ = ϕ and χ = η there are just two invariants for which we use the notation: Finally we turn to the dihedral group D 6 which is the symmetry group of regular hexagon.
It is isomorphic to D 3 × Z 2 , and therefore inherits the group theoretical structure discussed above, except that each representation carries an additional Z 2 charge, which is conserved in tensor decompositions.Thus we have four one-dimensional representations 1 + , 1 − , 1 + , 1 − (where 1 + denotes the total singlet) and two two-dimensional representations 2 + , 2 − .The decompositions of these representations follow from the D 3 ones, for example we have Thus in D 6 the tensor product (2 − ⊗ 2 − ⊗ 2 − ) does not contain a singlet.

B Fit Results
Parameter QLν D -

C Scalar Potential
In this section we consider an explicit scalar potential that generates the VEVs we have assumed in Section 3, serving merely as a proof of existence.
In addition to the scalars φ and χ we need to introduce a new (SM singlet) scalar ψ in order to break the U (1) symmetries in the scalar potential to a single continoous global symmetry that can be identified with U (1) F .The transformation properties under and the most general, renormalizable scalar potential for these fields is given by where the D 6 singlet contractions are explained in Appendix A and we take κ 1 , κ 2 , λ χψ and ρ to be real.The ground state of this potential is most easily studied in the limit when For a suitable range of parameters (see below), one can easily show that the ground state at leading order in ρ and κ 2 is given by There is a symmetry exchanging φ 1 ↔ φ 2 in the potential, which are connected by a D 6 transformation that we can use to assume the large VEV in the φ 1 direction without loss of generality.The VEVs of v 2 and v ψ only arise at O(κ) and O(κρ), respectively: In order to suppress the VEVs of φ 2 and ψ sufficiently, i.e. to ensure the validity of e.g.Eq. (2.1), we need roughly v 2 /v 1 ∼ v 2 /v χ ∼ κ 2 ε 2 χ ∼ 10 −4 .Such a small coupling is technically natural, since in the limit of κ 2 → 0, λ χψ → 0 (or ρ → 0) the Lagrangian acquires a larger symmetry.This can be seen from spelling out the third line of the potential explicitly: Indeed this part only breaks the additional U (1) 3 symmetry of the scalar kinetic terms (besides the remaining U (1) F ) if ρ = 0 and κ = 0 or λ χψ = 0.
This observation is also crucial to understand why the additional field ψ is needed: its coupling ρ is the only parameter that breaks the U (1) symmetry under which χ is neutral and φ 1 and φ 2 carry opposite charges.Moreover it makes clear that we expect (in addition to the massless U (1) F Goldstone) a very light pseudoscalar in the spectrum whose mass is suppressed by the small couplings κ 2 , λ χψ and ρ.
After these analytical considerations we finally provide a numerical example, taking the following set of parameters: Finally the scalar mass spectrum is given by one massless Goldstone, 6 massive scalars with masses {6.3, 6.3, 6.0, 6.0, 3.9, 2.6}m and a light scalar with mass 2.9 • 10 −5 m.Using the lower bound on Λ from Section 4.2 (corresponding to an axion mass m a 14 meV), we get a lower bound on m roughly given by m ∼ Λ/190 > 5 • 10 8 GeV, so the light scalar has a mass 15 TeV.

Table 1 :
The field content and U (2) F quantum numbers.

Table 2 :
Input values of quark and charged lepton Yukawas and quark mixing angles at 10

Table 3 :
Best fits in the quark and charged lepton sector.

Table 6 :
Best fits of the combined quark and lepton sector including CKM phase and Dirac neutrinos, with normal ordering (NO) or inverted ordering (IO).The complete set of parameters can be found in Table15.

Table 8 :
Range of predictions for m i and m β scanning over fits with Dirac Neutrino charges

Table 10 :
Best fits for the D 6× U (1) model including CKM phase and Majorana neutrinos.

Table 11 :
Predictions for neutrino masses and observables for the fits in Table10.Since the prediction for m ββ strongly depends on possible phases in the PMNS matrix, here we display the maximal possible value m max ββ .

Table 12 :
Range of predictions for m i , m β and m ββ scanning over fits with χ 2 < 20 and χ 2 O(1) < 20.The last column indicate the values preferred by most fits.