A Realistic U(2) Model of Flavor
CCERN-TH-2018-121TTP18-019
A Realistic U(2) Model of Flavor
Matthias Linster a and Robert Ziegler a,b a Institut f¨ur Theoretische Teilchenphysik, Karlsruhe Institute of Technology,Engesserstraße 7, 76128 Karlsruhe, Germany b Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland
Abstract
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 parametersthat control the U (2) breaking. In contrast to previous U (2) models this setup can berealized without supersymmetry and provides an excellent fit to all SM flavor observablesincluding neutrinos. We also consider a variant of this model based on a D × U (1) F flavor symmetry, which closely resembles the U (2) structure, but allows for Majorananeutrino masses from the Weinberg operator. Remarkably, in this case one naturallyobtains large mixing angles in the lepton sector from small mixing angles in the quarksector. The model also offers a natural option for addressing the Strong CP Problem andDark Matter by identifying the Goldstone boson of the U (1) F factor as the QCD axion. a r X i v : . [ h e p - ph ] A ug Introduction
One of the prominent problems of the Standard Model (SM) is the presence of large hierarchiesin fermion masses and mixings. Even neglecting neutrino masses, which might have a differentorigin, the Yukawa couplings span a range from 10 − for the electron up to unity for the topquark. Mixing angles in the quark sector are small and hierarchical, while all mixing anglesin the lepton sector are sizable. Explaining these hierarchies is referred to as the “SM FlavorPuzzle” (see e.g. Ref. [1] for a review).A popular framework to address this problem is in terms of approximate flavor (or hor-izontal) symmetries. The SM fermions are charged under this symmetry, so that most ofthe Yukawa couplings are forbidden in the symmetry limit. The flavor symmetry is sponta-neously broken by vacuum expectation values of scalar fields (the so-called flavons), whichallows to estimate the Yukawa couplings using a spurion analysis. Within an effective fieldtheory approach, appropriate powers of flavon insertions are needed to make a given Yukawaoperator invariant under the flavor symmetry, suppressed by some large UV cutoff scale. Theflavon VEVs are assumed to be slightly below this cutoff scale, so that SM Yukawas arisefrom powers of these small order parameters. The effective operators have coefficients thatare not predicted by the model, but should not be too large or small, in order to explain allhierarchies with the approximate flavor symmetry alone.While a plethora of this kind of models have been constructed (see e.g. Ref. [1] andreferences therein), a particularly 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 withan SO (10) GUT structure, and therefore viable only in a supersymmetric (SUSY) context (ormore 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 thenleads to three texture zeros in the quark mass matrices, which imply certain relations betweenCKM mixing angles and quark masses, in particular V ub /V cb = (cid:112) m u /m c . Unfortunately, thisprediction is incompatible with the current experimental precision of V ub and V cb , and thissimple 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 themodel predictions and comply with experimental data. In Ref. [5] a SUSY SO (10) modelwith a D × U (1) flavor symmetry was studied, which mimicked the original U (2) structurewith three texture zeros, but is also in conflict with present values of CKM elements. Amore recent study has been performed in Ref. [6], which has shown that the problematicrelation can be fixed by taking flavor quantum numbers compatible only with an SU (5) GUTstructure. This allows the presence of large rotations in the right-handed (RH) down sectorthat correct the predictions, as suggested in Ref. [4]. Relaxing the SO (10) structure admitsto consider also non-supersymmetric models, and in Ref. [7] such a model was constructedwith 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 cansuccessfully explain the observed deviations in R K [8] and R K ∗ [9] by the tree-level exchangeof a Z (cid:48) boson in the TeV range. In contrast to many Z (cid:48) models that address the anomalies,the couplings to fermions are related to the flavor sector and thus essentially predicted interms 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 due1o large mixing angles in the RH down sector, allowing for an excellent fit to CKM angles andquark 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 neutrinosector, which in this framework can be straightforwardly reproduced by adding three lightSM singlets with suitable U (2) quantum numbers and Dirac masses. The fit to the full SMfermion sector is excellent, and predicts the overall mass scale in the neutrino sector belowcurrent cosmological bounds. We further discuss a variant of the U (2) model where the SU (2)factor is replaced by the discrete group D . The breaking pattern and the resulting Yukawamatrices closely resemble the SU (2) case. The only difference is a flipped sign in the 1-2entry of the mass matrices, which has no effect in the quark and charged lepton sector, butallows to obtain Majorana neutrinos masses from the Weinberg operator. In contrast to theDirac case the parametric flavor suppression of the neutrino mass matrix is fixed purely bycharged lepton charges. Remarkably, this matrix is automatically anarchical, and thereforeallows for an excellent fit to neutrino data, again predicting the overall neutrino mass scalein 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 therole of the QCD axion and has (flavor-violating) couplings to fermions that are predicted bythe flavor model, in the spirit of Refs. [10–12]. In contrast to single U (1) flavor models, herethe 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 byastrophysics and not by precision flavor observables. It is well-known that the axion can bean excellent Dark Matter (DM) candidate for large ranges of the U (1) breaking scale, whichhere is directly connected to the UV cutoff of the flavor model. In this way the model offersa 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 anddiscuss the structure of the quark and charged lepton sector before addressing the (Dirac)neutrino sector. We then consider a D × 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 concludein Section 5. In three Appendices we provide more details on the group theoretical structureof D and D (cid:39) D × Z , include more details about the numerical fit, and discuss an explicitexample of the scalar potential generating the flavon VEVs in the D × U (1) model. U (2) Model of Flavor
In this section we define our framework and show how hierarchies in the quark and chargedlepton sector arise from the U (2) flavor symmetry. After discussing the analytical relationsbetween 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 thenumerical fit. We conclude this section with a general discussion of the flavor structure ofneutrino masses, motivating the D × U (1) flavor model in the next section. We consider an extension of the SM with a global flavor symmetry group U (2) F . Locallythis group is isomorphic to SU (2) F × U (1) F , under which SM fermions are charged. This2ymmetry group is assumed to be broken slightly below a UV scale Λ, which sets the relevantmass scale for additional dynamics. We also assume that the scale Λ is large enough to safelyneglect the impact of these new degrees of freedom on phenomenology. Thus, we simply workwith an effective theory with cut-off scale Λ that only involves SM fields and spurions thatparametrize 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) GUTstructure, i.e. they are specified by the quantum number of the two SU (5) representations = Q, U, E and = 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 fourcharges { X a , X a , X , X } for { a , a , , } with a = 1 ,
2. It turns out that a suc-cessful fit to the observed fermion masses and mixings can be achieved for the following simplechoice for U (1) F charges: X = 0 , X a = X a = X = 1 . (2.1)The breaking of the flavor symmetry is described by two scalar spurions φ and χ , whichtransform under U (2) F as φ = − and χ = − . These fields acquire the following vacuumexpectation values (VEVs): (cid:104) φ (cid:105) = (cid:18) ε φ Λ0 (cid:19) , (cid:104) χ (cid:105) = ε χ Λ , (2.2)where we will take ε φ ∼ ε χ ∼ O (0 . U (2) F , a a H φ a χSU (2) F U (1) F − − Table 1:
The field content and U (2) F quantum numbers. Yukawa couplings require additional spurion insertions in order to be U (2) F -invariant. Thisleads 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 L u = λ u Λ χ ( φ ∗ a Q a )( φ ∗ b U b ) H + λ u Λ χ (cid:15) ab Q a U b H + λ u Λ χ ( φ ∗ a Q a ) U H + λ u Λ ( (cid:15) ab φ a Q b )( (cid:15) cd φ c U d ) H + λ u Λ ( (cid:15) ab φ a Q b ) U H + λ u Λ χ Q ( φ ∗ a U a ) H + λ u Λ Q ( (cid:15) ab φ a U b ) H + λ u Q U H , (2.3)and similar in the down and charged lepton sector. After inserting the spurion VEVs the cutoffdependence 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 as3 yuk = Q T Y u U H + · · · ) the result Y u ≈ λ u ε φ ε χ λ u ε χ λ u ε φ ε χ − λ u ε χ λ u ε φ λ u ε φ λ u ε φ ε χ λ u ε φ λ u , Y d ≈ λ d ε φ ε χ λ d ε χ λ d ε φ ε χ − λ d ε χ λ d ε φ λ d ε φ ε χ λ d ε φ ε χ λ d ε φ λ d ε χ , (2.4) Y e ≈ λ e ε φ ε χ λ e ε χ λ e ε φ ε χ − λ e ε χ λ e ε φ λ e ε φ λ e ε φ ε χ λ e ε φ ε χ λ e ε χ , where λ fij are (in general complex) O (1) coefficients and we have kept only the leading con-tributions in ε φ,χ . Note that, in contrast to the supersymmetric U (2) model in Ref. [6], thereare no holomorphy constraints, which leads to a more general Yukawa pattern.One can show that the λ , λ , λ entries give only subleading corrections to quark massesand mixings, which are relatively suppressed by at least ε φ . Thus, effectively, three texturezeros appear in the Yukawa matrix, much as in the supersymmetric models [6], and to goodapproximation we obtain the Yukawa couplings Y u ≈ λ u ε χ − λ u ε χ λ u ε φ λ u ε φ λ u ε φ λ u , Y d ≈ λ d ε χ − λ d ε χ λ d ε φ λ d ε φ ε χ λ d ε φ λ d ε χ ,Y e ≈ λ e ε χ − λ e ε χ λ e ε φ λ e ε φ λ e ε φ ε χ λ e ε χ . (2.5)Because of the hierarchical structure and the presence of the texture zeros, it is possibleto analytically derive some approximate results for the singular values and the rotationsto the mass basis [7]. One can also perturbatively diagonalize the Yukawa matrices, andobtain the following estimates for singular values and CKM matrix elements (neglecting O (1)coefficients): y u ∼ ε χ /ε φ , y d ∼ y e ∼ ε χ /ε φ , V ub ∼ ε χ /ε φ ,y c ∼ ε φ , y s ∼ y µ ∼ ε φ ε χ / (cid:113) ε φ + ε χ , V cb ∼ ε φ ,y t ∼ , y b ∼ y τ ∼ (cid:113) ε φ + ε χ , V us ∼ ε χ /ε φ . (2.6)These expressions can be compared to the (1 σ ) ranges for fermion mass ratios and CKMelements, taken for definiteness at 10 TeV m u m t ≈ λ (7 . ÷ . , m d m b ≈ λ (4 . ÷ . , m e m τ ≈ λ . , V ub ≈ λ m c m t ≈ λ . , m s m b ≈ λ (2 . ÷ . , m µ m τ ≈ λ . , V cb ≈ λ , (2.7)where λ = 0 . ≈ V us and y b (10 TeV) ≈ λ . , y τ (10 TeV) ≈ λ . . Within roughly a factor λ ,all hierarchies can be reproduced taking ε φ ∼ V cb ∼ λ , ε χ ∼ λ ÷ , (2.8)4nd therefore a good fit to masses and mixings can be expected with input parameters λ fij thatare indeed O (1). Moreover, it is clear that there must be four relations in each fermion sectorbetween the 3 singular values and the 3+3 rotation angles. For real h fij it is straightforwardto work out these predictions exactly [7] and expand the result in ratios of the hierarchicaleigenvalues. One can then relate the 1-2 and 1-3 rotations in the left- and right-handed sectorsto the 2-3 rotations and the eigenvalues. With the convention Y = V L Y diag V † R , V L = V L V L V L , V R = V R V R V R , (2.9)where V ij are orthogonal rotation matrices in the i - j plane that are parametrized by theangles s ij ≡ sin θ ij , one obtains up to percent corrections s Lu ≈ − s Ru ≈ (cid:114) m u m c , s Lu ≈ − s Lu s Lu , s Ru ≈ s Ru s Lu ,s Ld ≈ − s Rd ≈ (cid:114) m d m s (cid:113) c Rd , s Ld ≈ − s Ld s Ld (cid:18) − s Rd c Rd s Ld m s m b (cid:19) , s Rd ≈ s Rd c Rd s Ld ,s Re ≈ − s Le ≈ (cid:114) m e m µ (cid:113) c Le , s Re ≈ − s Re s Re (cid:18) − s Le c Le s Re m µ m τ (cid:19) , s Le ≈ s Le c Le s Re , (2.10)where 2-3 rotations angles are large in the RH down and LH charged lepton sector, andCKM-like in all other sectors s Rd ∼ s Le ∼ , s Lu ∼ s Ru ∼ s Ld ∼ s Re ∼ V cb . (2.11)One therefore obtains for the CKM elements (in our conventions V CKM = V uTL V d ∗ L ) thepredictions | V ub | ≈ (cid:12)(cid:12)(cid:12)(cid:12)(cid:114) m u m c | V cb | − e iφ (cid:114) m d m s (cid:113) c Rd s Rd c Rd m s m b (cid:12)(cid:12)(cid:12)(cid:12) , | V td | ≈ (cid:114) m d m s (cid:113) c Rd (cid:12)(cid:12)(cid:12)(cid:12) | V cb | − e iφ s Rd c Rd m s m b (cid:12)(cid:12)(cid:12)(cid:12) , | V us | ≈ | s Ld − s Lu | ≈ (cid:12)(cid:12)(cid:12)(cid:12)(cid:114) m d m s (cid:113) c Rd − e i ( φ − φ ) (cid:114) m u m c (cid:12)(cid:12)(cid:12)(cid:12) , | V cb | ≈ | V ts | ≈ | s Ld − s Lu | , (2.12)where we included also relative phases φ , , see Ref. [6] for details. In the original U (2) modelsin Ref. [2, 3], the rotation angle in 2-3 RH down sector s Rd was taken to be of the order ofthe other 2-3 rotation angles, s Rd ∼ V cb . From the above equations, this directly leads tothe accurate prediction | V ub /V cb | ≈ (cid:112) m u /m c which deviates from experimental data by morethan 3 σ . This is the reason why here this angle is taken to be large, s Rd ∼ c Rd ∼ / √
2, whichthen allows to obtain an excellent fit to CKM angles as we demonstrate in the next section(see also Refs. [4, 6, 7]).
We now perform a numerical fit to the model parameter set { λ u,d,eij , ε φ , ε χ } . For simplicity,we restrict to real λ u,d,eij and demonstrate later on that the CKM phase can be obtained bytaking a complex parameter λ u . The experimental input parameters are therefore the quarkand charged lepton masses and the CKM mixing angles. For concreteness we take them inthe MS scheme at 10 TeV from Ref. [13], with a symmetrized 1 σ error taken to be the largerone. All input parameters are summarized in Table 2.5uantity Value y u (5 . ± . × − y d (1 . ± . × − y s (2 . ± . × − y c (2 . ± . × − y b (1 . ± . × − y t . ± . y e (2 . ± . × − y µ (6 . ± . × − y τ (1 . ± . × − θ . ± . θ (4 . ± . × − θ (3 . ± . × − Table 2:
Input values of quark and charged lepton Yukawas and quark mixing angles at 10TeV taken from Ref. [13].
The quality of the fit with a given model parameter set { λ u,d,eij , ε φ , ε χ } is measured bytwo functions χ and χ O (1) . The first quantity is the usual χ that indicates how well theexperimental input values are reproduced by the fit. It is obtained by plugging the modelparameters into the Yukawa matrices in Eq. (2.1) and calculating numerically the singularvalues y q,l and the CKM mixing angles θ ij in the PDG parametrization. These values areused with the experimental input above to obtain χ defined as χ = (cid:88) q = u,d,s,c,b,t ( y q − y q, exp ) ( σy q, exp ) + (cid:88) (cid:96) = e,µ,τ ( y (cid:96) − y (cid:96), exp ) ( σy (cid:96), exp ) + (cid:88) ( ij )=(12) , (13) , (23) ( θ ij − θ ij, exp ) ( σθ ij, exp ) . (2.13)In order to explain Yukawa hierarchies solely by U (2) F breaking, the parameters λ u,d,eij shouldbe O (1). The meaning of this requirement is somewhat fuzzy, and here we choose to quantifyit by introducing a measure χ O (1) defined as χ O (1) = (cid:88) λ pij (cid:16) log( | λ pij | ) (cid:17) · . , (2.14)where i, j = 1 , , p = u, d, e . This corresponds to the assumption that the λ u,d,eij 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,eij lie with a probability of 95 % within the interval[1 / , χ O (1) of a single parameter λ = { , , , , , } (or the inverse) is ∆ χ O (1) = { , , , , , } . We consider a fit satisfactory as long as χ O (1) ≤ χ and χ 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 , χ O (1) as defined above, along with thesmallest and largest | λ u,d,eij | . For the fit QL1 R we have minimized χ + χ O (1) , while for QL2 R
6e have minimized χ O (1) while keeping χ ≤ χ (QL3 R ).Fit ε φ ε χ min | λ u,d,(cid:96)ij | max | λ u,d,(cid:96)ij | χ χ O (1) QL1 R R QL3 R Table 3:
Best fits in the quark and charged lepton sector.
Indeed there are enough free parameters to obtain a perfect fit to observables, however oneneeds χ O (1) as large as 35 and O (1) parameters as small as ≈ /
9, so this fit should bediscarded according to our quality requirement χ 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 ofthe naive size estimated in Eq. (2.8).Finally we demonstrate that the CKM phase δ CP can be easily included. For simplicitywe restrict to the case where only the 33 entry in the up-quark Yukawa matrix is complex,i.e. λ u → λ u e iδ . In a realistic setup where all Yukawas have phases, the fit can only getbetter. In the χ 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 . ± . . Including δ leads to even better fits (QL1 and QL2), which we show in Table 4. Thisdemonstrates that an excellent fit for quark and charged lepton sector, including the CKMphase, can be obtained with all O (1) parameters lying between 1 / . . ε φ ε χ min | λ u,d,(cid:96)ij | max | λ u,d,(cid:96)ij | χ χ O (1) QL1 0.025 0.009 1/2.9 2.1 0.6 5.8QL2 0.024 0.008 1/2.8 1.9 13 4.8
Table 4:
Best fits in the quark and charged lepton sector including the CKM phase.
In the neutrino sector we have to distinguish whether neutrinos are Dirac or Majorana. Webegin with the discussion of the Dirac scenario, since the Majorana case in the U (2) F model isstrongly disfavored as we will discuss below. To this extent we introduce SM singlets N a , N with U (1) F charges X Na and X N , where N a transforms as a doublet of SU (2) F and N as asinglet. The Lagrangian then allows for a Yukawa coupling L ν = L T Y ν N H (we assume thatthe Majorana mass term is forbidden, e.g. by exact lepton number conservation). As in thecharged lepton sector, one can obtain its structure from a spurion analysis as Y ν = λ ν ε φ ε | X Na | χ λ ν ε | X Na | χ λ ν ε φ ε | X N | χ − λ ν ε | X Na | χ λ ν ε φ ε | X Na − | χ λ ν ε φ ε | X N | χ λ ν ε φ ε | X Na | χ λ ν ε φ ε | X Na | χ λ ν ε | X N | χ . (2.15)7t is clear that in order to obtain sub-eV neutrinos one needs large U (1) F charges X Na, > , (13) , (31) entries to masses and mixings are again sub-leading, and we can drop them asin the previous section and are left with the Dirac neutrino mass matrix m Dν ≈ v λ ν ε X Na χ − λ ν ε X Na χ λ ν ε φ ε X Na − χ λ ν ε φ ε X N χ λ ν ε φ ε X Na χ λ ν ε X N χ . (2.16)It is well-known that an anarchical neutrino mass matrix can give a good fit to neutrinoobservables, which can be achieved taking X Na = X N (since ε χ ∼ ε φ ), giving m Dν ≈ v ε X Na − χ λ ν ε χ − λ ν ε χ λ ν ε φ λ ν ε φ ε χ λ ν ε φ ε χ λ ν ε χ . (2.17)In order to obtain an overall neutrino mass scale (cid:46) . X N (cid:38)
5, so that tinyneutrino masses arise from somewhat large U (1) F charges and the smallness of the U (2) F breaking parameters, ε χ,φ ∼ . Normal Ordering (NO)
Quantity Value∆ m (7 . ± . × − ∆ m (2 . ± . × − sin θ . ± . θ . ± . θ . ± . Inverted Ordering (IO)
Quantity Value∆ m (7 . ± . × − ∆ m ( − . ± . × − sin θ . ± . θ . ± . θ . ± . Table 5:
Experimental values of neutrino mass differences and PMNS mixing angles fornormal (NO) and inverted hierarchy (IO), taken from NuFIT 3.2 (2018) [14, 15].
We then plug the neutrino model parameters λ νij for fixed charges X Na , X N into the Yukawamatrices in Eq. (2.17) and calculate numerically the singular values and the PMNS mixingangles θ ij in the standard parametrization. To the χ defined in Eq. (2.13) we add thecorresponding expression χ ν in the neutrino sector χ ν = (cid:88) ( ij )=21 , / (∆ m ij − ∆ m ij, exp ) ( σ ∆ m ij, exp ) + (cid:88) ( ij )=(12) , (13) , (23) (sin θ ij − sin θ ij, exp ) ( σ sin θ ij, exp ) , (2.18)and similarly we include the coefficients λ νij in the measure χ O (1) defined in Eq. (2.14). Wethen perform a simultaneous fit to quark, charged lepton and neutrino sector including a For the angle sin θ we actually use the full χ function provided by the NuFIT collaboration instead ofassuming the Gaussian error in Table 5. λ u as discussed in the last section (for simplicity we omit phases in the neutrinosector, including them would make the fit only better). The fit results are shown in Table 6,both for NO and IO.Fit X Na X N ε φ ε χ min | λ u,d,e,νij | max | λ u,d,e,νij | χ χ O (1) QL ν D -1 (NO) 6 6 0.026 0.012 1 / . ν D -2 (NO) 6 6 0.024 0.013 1 / . ν D -3 (NO) 5 5 0.022 0.006 1 / . ν D -4 (NO) 5 5 0.021 0.006 1 / . QL ν D (IO) 6 6 0.015 0.013 1 / . Table 6:
Best fits of the combined quark and lepton sector including CKM phase and Diracneutrinos, with normal ordering (NO) or inverted ordering (IO). The complete set of parame-ters can be found in Table 15.
As expected, good fits are obtained only for equal charges X Na = X N = 5 ÷
6. There is clearlya strong preference for NO, as can be seen in both quality parameters χ and χ O (1) (and thesmallest/largest λ ij ). According to our quality requirement χ O (1) <
20, we should actuallydiscard the IO possibility, since all fits with inverted mass ordering violate this criterion, andwe include it just for illustrative purposes.Comparing to the fit of quark and charged lepton sector only (cf. Table 4), one can seethat including neutrinos makes the fits slightly worse, but still with O (1) coefficients between1 / m i as probed bysatellite telescopes, and the effective neutrino mass m β = (cid:113)(cid:80) i m i | U ei | as measured in the β -decay spectrum close to the endpoint. All predictions are summarized in Table 7.Fit m [meV] m [meV] m [meV] (cid:80) m i [meV] m β [meV]QL ν D -1 0.5 8.6 50 59 9QL ν D -2 4.6 9.6 50 64 10QL ν D -3 0.4 8.6 50 59 9QL ν D -4 0.4 8.6 50 59 9 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 yetmeasured, we also give a range for these predictions scanning over many fits with X Na = X N = 5 , χ <
20 andthe quality requirement χ O (1) <
20 (which excludes IO). In this way we obtain predictions forthe ranges of (cid:80) m i and m β as shown in Table 8, where we also indicate the value preferredin most fits. 9uantity Range [meV] Preferred values [meV] (cid:80) m i
58 – 110 60 – 65 m β Table 8:
Range of predictions for (cid:80) m i and m β scanning over fits with Dirac Neutrino charges X Na = X N = 5 , χ <
20 and χ O (1) <
20. In brackets indicated are the values preferredby most fits.
We notice that the predicted range for m β is an order of magnitude below the expectedfuture sensitivity of m β (cid:46) . (cid:80) m i < .
12 eV [17]and in the reach of the EUCLID satellite that is expected to measure (cid:80) m i with an error ofabout 0 .
05 eV [18, 19]. Note that the lower bound on the predicted range of (cid:80) m i essentiallysaturates the minimal value that is obtained for a massless lightest neutrino, which (including1 σ errors) is given by 58 meV for normal ordering.Finally, we discuss the case of Majorana Neutrinos. In addition to the neutrino Yukawacoupling, the Lagrangian contains a Majorana mass term, L ν = L T Y ν N H +1 / N T M ν N +h . c . The Yukawa matrix Y ν is the same as in Eq. (2.15), while the Majorana mass matrix can beobtained as M ν = M κ ε φ ε | X Na | χ κ ε φ ε | X Na | χ κ ε φ ε | X Na + X N | χ κ ε φ ε | X Na | χ κ ε φ ε | X Na − | χ κ ε φ ε | X Na + X N − | χ κ ε φ ε | X Na + X N | χ κ ε φ ε | X Na + X N − | χ κ ε | X N | χ , (2.19)where we factored out a single mass scale M that is taken of the order of the usual see-sawscale, M ∼ GeV. One can therefore integrate out the heavy singlets and get light neutrinomasses from the Weinberg operator y ij /M ( L i H )( L j H ), according to the type-I seesaw formula m Mν = v Y ν M − ν Y Tν . (2.20)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 φ suppression. It turns outthat this extra suppression spoils the naive EFT spurion analysis of the Weinberg operatorusing only the charges of L a , L (since negative powers of φ appear in the UV theory), andone has to use Eq. (2.20) to calculate m Mν . We first assume that X NA ≥ X N ≥
0, sothat one can drop the absolute values and obtain for the parametric structure of the lightneutrino mass matrix m Mν ∼ v M ε χ /ε φ ε χ /ε φ ε χ /ε φ ε χ /ε φ /ε φ ε χ /ε φ ε χ /ε φ ε χ /ε φ ε χ ∼ ε ε /ε ε ε , (2.21)where ε ∼ ε φ ∼ ε χ (notice that the charges X Na, drop out). Such a structure is clearly ruledout, since it gives singular values { ε , ε , /ε } , which would imply normal hierarchy alongwith a parametric prediction for the ratio of mass differences ∆ m / ∆ m ∼ ε × ε thatis way too small. Moreover, one can check that also different charge assignments for X Na, do not allow to obtain a Majorana neutrino mass matrix that leads to a good fit, besides10osing predictivity. Indeed the main theoretical advantage of Majorana neutrinos over Diracneutrinos would be a scenario in which the effective Majorana mass matrix does not dependon the details of the UV physics, i.e. the choice of X Na, .We conclude this section with the observation that the Majorana scenario would workperfectly 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 ε | X Na | χ , and X Na ≥ X N ≥
0, the effective lightneutrino mass matrix would be given by (the dependence on X Na, drops out again) m Mν ∼ v M ε χ ε φ ε χ ε χ ε φ ε χ ε φ ε χ ε φ ε χ ε φ ε χ ε φ ε χ ∼ ε ε ε ε ε ε ε ε ε , (2.22)which apart from the subleading 11 , ,
31 entries has only very mild ε χ /ε φ hierarchies andsuggests a very good fit to neutrino observables. Note this absence of hierarchies is actuallya prediction of the quark and charged lepton sector, which requires equal charges for the left-handed doublets L a and L , and order parameters of similar size, ε χ ∼ ε φ . If therefore the1-2 elements were symmetric instead of anti-symmetric, all low-energy mass matrices wouldfollow the same hierarchical pattern, differing only in the U (1) F charge assignment of thethird generation, which is 0 for Q , U , E and 1 for D , L . Thus the light 2 × ε χ , giving m { u,d,e,ν } ∼ ε ε ε { ε, ε , ε, ε } { ε, ε, ε , ε } { , ε, ε, ε } , (2.23)where we neglected the mild ε χ /ε φ hierarchy that is responsible for e.g. the Cabibbo angle. Aswe discuss in the following section, this simple pattern allows for an excellent fit to all fermionobservables, and the necessary 1-2 symmetric structure can be obtained when considering the(discrete) dihedral group D instead of SU (2) as flavor symmetry, which closely resemblesthe SU (2) structure apart from a sign flip in the 1-2 entries. D × U (1) Model of Flavor
In this section we consider the same framework with a D × U (1) flavor symmetry, whichclosely resembles the U (2) case. We first introduce some D (cid:39) D × Z group theory anddiscuss the resulting flavor structure of quark and charged lepton masses, as well as theWeinberg operator. After some brief analytical considerations for the resulting predictionsfor neutrino observables, we perform a numerical fit to all fermion observables and concludewith a discussion of the phenomenological implications. As we have just discussed, we want to mimic the structure of U (2) within a discrete flavorgroup that allows for a symmetric singlet contraction of two doublets. The simplest such groupis the dihedral group D , the symmetry group of an equilateral triangle, which is discussed indetail in Appendix A. This group is actually a subgroup of SO (3) and not of its double cover11 U (2), and it is isomorphic to the permutation group S . It features two one-dimensionalrepresentations and (cid:48) and one two-dimensional representation . The contraction of twodoublets ψ = ( ψ , ψ ) and φ = ( φ , φ ) into the singlet is given by( ψ ⊗ φ ) = ψ φ + ψ φ . (3.1)Therefore we could simply assign the SM and spurion fields to D representations that followthe SU (2) ones, i.e. the doublets a , a , φ a are in a of D and all other fields are totalsinglets. 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( ψ ⊗ φ ⊗ χ ) = ψ φ χ + ψ φ χ . (3.2)This implies that in contrast to the SU (2) model a large 1-1 entry is generated, for examplein the up-sector by the operator L ⊃ ( φ ⊗ Q a ⊗ U a ) Hχ = 1Λ ( φ Q U + φ Q U ) Hχ = ε φ ε χ Q U H , (3.3)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 sucha contraction is forbidden by the Z center of SU (2), under which the doublets are oddand the singlet is even. Therefore, we consider D × Z which is isomorphic to D , thesymmetry group of a regular hexagon (see Appendix A for details), and finally make thecharge assignment as in Table 9. The additional Z factor ensures that the contraction of a a H φ a χD × Z − − + + + − + U (1) F − − Table 9:
The field content and ( D (cid:39) D × Z ) × U (1) F quantum numbers. three − doublets does not contain the total singlet + , and in the quark and charged leptonsector we obtain the very same spurion analysis as for U (2) in Section 2 (see Eq. (2.4)), exceptfor the sign in the 1-2 entry: Y u ≈ λ u ε φ ε χ λ u ε χ λ u ε φ ε χ λ u ε χ λ u ε φ λ u ε φ λ u ε φ ε χ λ u ε φ λ u , Y d ≈ λ d ε φ ε χ λ d ε χ λ d ε φ ε χ λ d ε χ λ d ε φ λ d ε φ ε χ λ d ε φ ε χ λ d ε φ λ d ε χ , (3.4) Y e ≈ λ e ε φ ε χ λ e ε χ λ e ε φ ε χ λ e ε χ λ e ε φ λ e ε φ λ e ε φ ε χ λ e ε φ ε χ λ e ε χ . In the neutrino sector we work with the effective Weinberg operator y ij /M ( L i H )( L j H ), whichcan be induced by the type-I seesaw mechanism as discussed in the previous section. Its Note we cannot use the double cover ˜ D (which is an actual subgroup of SU (2)) for this purpose, sincethat doublet x that contains no singlet in its cubic contraction, contains the singlet in its antisymmetricquadratic contraction. D × U (1) F quantum numbers of the chargedleptons, which gives for the light Majorana neutrino mass matrix m ν ≈ v M λ ν ε χ ε φ λ ν ε χ λ ν ε φ ε χ λ ν ε χ λ ν ε φ λ ν ε φ ε χ λ ν ε φ ε χ λ ν ε φ ε χ λ ν ε χ . (3.5)Here we have used the same vacuum expectation values as before (cid:104) φ (cid:105) = (cid:18) ε φ Λ0 (cid:19) , (cid:104) χ (cid:105) = ε χ Λ , (3.6)although in contrast to the SU (2) F case we cannot use D transformations in order to assumethis VEV for φ without loss of generality. Therefore, we provide an explicit scalar potentialin Appendix C with only one additional scalar field that generates dynamically the aboveVEVs . Altogether, we obtain to good approximation the mass matrices m u ≈ v λ u ε χ λ u ε χ λ u ε φ λ u ε φ λ u ε φ λ u , m d ≈ v λ d ε χ λ d ε χ λ d ε φ λ d ε φ ε χ λ d ε φ λ d ε χ , (3.7) m e ≈ v λ e ε χ λ e ε χ λ e ε φ λ e ε φ λ e ε φ ε χ λ e ε χ , m ν ≈ v M λ ν ε χ λ ν ε χ λ ν ε φ λ ν ε φ ε χ λ ν ε φ ε χ λ ν ε χ . As discussed in the previous section, this model has the remarkable feature that the hierarchiesin the quark and charged lepton sector require ε φ ∼ ε χ , and therefore naturally gives rise toan approximately anarchic neutrino mass matrix with generically large mixing angles.Before we perform a numerical fit, we proceed with some analytical considerations. Inthe quark and charged lepton sector the analysis of the previous section is unaltered, sincethe flipped sign in the 1-2 entry does not play a role at leading order. In the neutrino sectorwe have 4 real parameters, which will enter the PMNS matrix together with three chargedlepton rotations angles controlled by a single free real parameter s Le , see Eq. (2.10). Theseparameters correspond to 5 observables (3 PMNS angles + 2 squared mass differences), soup to phases all parameters are fixed and one can predict the absolute neutrino mass scalesand related observables. There are 4 phases in the neutrino sector and 2 phases in the left-handed charged lepton rotations, which combine to 3 physical phases, one Dirac and twoMajorana phases. To study the prediction of the overall neutrino mass scale, we parametrizethe neutrino mixing matrix V ν (defined by V Tν m ν V = m diag ν ) in the standard CKM formmultiplied with a phase matrix P ν = diag( e iα , e iα ,
1) from the right and a phase matrix P (cid:48) from the left. Inverting the defining equation, we get from the vanishing 11 and 13 entriesthe two equations c ,ν m m e − i ( α + δ ν ) + s ,ν m m e − i ( α + δ ν ) + s ,ν c ,ν = 0 , (3.8) m m e − i (2 α + δ ν ) − m m e − i (2 α + δ ν ) + s ,ν c ,ν c ,ν c ,ν s ,ν s ,ν = 0 . (3.9) Also a tiny VEV along the lower component of φ is generated, which however is small enough to give onlynegligible contributions to masses and mixings. (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − c ,ν s ,ν m m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ s ,ν s ,ν c ,ν m m ≤ c ,ν s ,ν m m , (3.10)1 − m m ≤ s ,ν c ,ν c ,ν c ,ν s ,ν s ,ν m m ≤ m m . (3.11)The angles in the neutrino sector s ij,ν are connected to the observed PMNS mixing anglesthrough V PMNS = ( V eL ) T V ν . Since the 1-2 rotation in the charged lepton sector is small, ∼ (cid:112) m e /m µ ≈ .
07, we have to good approximation s ,ν ≈ s , but 2-3 rotations in thecharged lepton sector are large, so that both θ and θ generically receive large contributionsfrom the charged lepton sector. Nevertheless one can easily verify that Eq. (3.10) cannot besatisfied for inverted mass ordering, while for normal ordering one can obtain an upper boundon the lightest neutrino mass m , by maximizing the neutrino mixing angles s ,ν and s ,ν with a suitable choice of phases. If one neglects the charged lepton contribution to s , one canshow that m ≤
11 meV, which in turn leads to upper bounds (cid:80) m i ≤
76 meV, m β ≤
14 meVand m ββ ≤
13 meV. This estimate is confirmed by the numerical analysis in the next section.
We now perform a simultaneous fit to quark, charged lepton and neutrino sector includinga phase in λ u 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, andinclude also the effective suppression scale M of Weinberg operator, which is of the order of10 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 ε φ ε χ min | λ u , d ,(cid:96)ij | max | λ u , d ,(cid:96)ij | χ χ O (1) M [10 GeV]QL ν M -1 0.025 0.009 1 / . ν M -2 0.024 0.009 1 / . Table 10:
Best fits for the D × U (1) model including CKM phase and Majorana neutrinos.The complete set of parameters can be found in Table 15. The corresponding predictions for the neutrino masses m i , its sum (cid:80) m i , the neutrinomass m β and the “effective Majorana mass” m ββ = (cid:12)(cid:12)(cid:80) U ei m i (cid:12)(cid:12) measured in neutrinolessdouble-beta 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 (cid:80) m i and m β are similar to the ones in the Dirac Neutrino case (cf. Table 7), while the effectiveMajorana mass is well below the expected sensitivities even in near future neutrinoless double-beta decay experiments [20]. 14it m [meV] m [meV] m [meV] (cid:80) m i [meV] m β [meV] m max ββ [meV]QL ν M -1 1 . . . . ν M -2 1 . . . . Table 11:
Predictions for neutrino masses and observables for the fits in Table 10. Since theprediction for m ββ strongly depends on possible phases in the PMNS matrix, here we displaythe maximal possible value m max ββ . Finally, we also give a range for the observables scanning over many fits on which we onlyimpose that χ <
20 and χ O (1) <
20. In this way we obtain predictions for (cid:80) m i , m β and m ββ lying in the ranges shown in Table 12, where we also indicate the value preferred inmost fits. This result agrees well with our estimate in the last section, where we have alsoincluded phases, so we expect the upper bounds on the mass scales to be approximately valideven when including phases in the numerical fit (the lower bounds again saturate the limitobtained from taking the lightest neutrino massless).Quantity Range [meV] Preferred values [meV] (cid:80) m i
59 – 78 60, 70 m β m max ββ Table 12:
Range of predictions for (cid:80) m i , m β and m ββ scanning over fits with χ <
20 and χ O (1) <
20. The last column indicates the values preferred by most fits.
We conclude this section with a discussion of the phenomenological implications of ourmodel. As we have seen, the flavor sector itself gives rise to quite narrow predictions forobservables in the neutrino sector, which are however far below the present experimentalsensitivities. In order to obtain other experimental signals, we have to rely on new low-energydynamics besides the SM. The natural candidate for such new degrees of freedom are the fieldsat the cut-off scale Λ, which we have not specified so far (in particular the radial componentsof the flavons φ and χ naturally get a mass at that scale). However, effects of these fields andother dynamics related to the UV completion are suppressed by powers of 1 / Λ, and there isno reason that Λ is sufficiently close to the electroweak scale in order to give rise to sizabledeviations from the SM. Still, it would be interesting to consider an explicit UV completionof the present model to study the structure of these effects in detail.Another option for light dynamics, which is essentially model-independent and well-motivated, is provided by the pseudo-scalars in the flavon fields. If there is no explicit breakingof the U (2) F symmetry, the associated Goldstone bosons are exactly massless, apart from alinear combination that can be identified with the QCD axion, which solves the strong CPproblem and gets a mass from non-perturbative effects. The easiest way to get rid of theorthogonal massless Goldstones is replacing SU (2) F by a discrete subgroup, which is anotheradvantage of the D × U (1) model discussed in this section. In this case there a single Gold-stone boson associated with the U (1) F factor that can naturally serve as the QCD axion, aswe are going to discuss in the next section. 15 The U (2) Axiflavon
As originally proposed in Ref. [10], a Goldstone boson arising from the breaking of globalflavor 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 flavormodel where the determinants of up-down and down-quark mass matrices are controlleddominantly by the U (1) F symmetry factor. In the present model this is indeed the case asdet m u ∼ ε χ and det m d ∼ ε χ , due to the presence of the approximate texture zeros, seeEq. (3.8). Moreover, if also the determinant of the charged lepton mass matrix depends onlyon the U (1) F breaking, the ratio of electromagnetic and color anomaly coefficients E/N isexpected to be a rational number close to 8 / U (1) F chargeassignment is actually compatible with SU (5), so it is clear that we get exactly E/N = 8 / U (1) F charges, while flavor-violating couplings to quarks and leptons arecontrolled by the unitary rotations that diagonalize the Yukawa matrices. Their parametricsuppression is determined by the U (2) F quantum numbers, and their numerical value by thefit to fermion masses and mixings. We then study the phenomenology of this axion, findingthat 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. We begin by identifying the axiflavon as the Goldstone boson arising from the spontaneousbreaking of U (1) F induced by the VEVs of φ and χ . In general, the Goldstone is a linearcombination of the phases a i of the scalar fields φ i with charge X i and (real) VEV V i , givenby a = (cid:88) i X i V i a i (cid:113)(cid:80) X j V j . (4.1)Thus, we find that χ and φ contain the Goldstone as (we ignore the radial mode) χ = ε χ Λ e − ia ( x ) / √ V , φ = (cid:18) ε φ Λ0 (cid:19) e − ia ( x ) / √ V , (4.2)where we have defined the U (1) F breaking scale V ≡ (cid:113) ε χ + ε 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 theanalogous terms in the down- and charged lepton sector. It is then convenient to change fieldbasis by performing a U (1) F transformation of the fermion fields f → f e iX f a ( x ) / √ V , (4.3)16hich will remove the a ( x ) dependence from the Yukawa sector, because of U (1) F invari-ance. Since this transformation is anomalous, it will generate axion couplings to gauge fieldstrengths, and since it is local it will modify fermion kinetic terms. The resulting couplingsto gluon and photon fields strengths are given by L anom = N a ( x ) √ V α s π G µν ˜ G µν + E a ( x ) √ V α em π F µν ˜ F µν , (4.4)with the dual field strength ˜ F µν = ε µνρσ F ρσ and the anomaly coefficients N = 12 (cid:0) X a + 2 X + 2 X a + X + 2 X a + X (cid:1) = 9 / , (4.5) E = 53 (2 X a + X ) + 43 (2 X a + X ) + 13 (cid:0) X a + X (cid:1) + (cid:0) X a + X (cid:1) + (2 X a + X ) = 12 . (4.6)Thus, we obtain E/N = 8 / U (1) F charge assignment is compatible with SU (5). The modification of fermion kinetic terms leadsto axion-fermion couplings in the flavor interaction basis L a = − ∂ µ a √ V (cid:88) f f † i σ µ X f i f i . (4.7)In the mass basis, defined as m f = V fL m diag f ( V fR ) † we have L a = − ∂ µ a √ V (cid:88) f = u,d,e (cid:104) g Lf i f j f † i σ µ f j + g Rf i f j f c † i σ µ f ci (cid:105) , (4.8)with g Lf i f j = ( V fL ) ki X f k ( V fL ) ∗ kj = X f a δ ij + ( X f − X f a )( V fL ) i ( V fL ) ∗ j , (4.9) g Rf i f j = ( V fR ) ∗ ki X f ck ( V fR ) kj = X f ca δ ij + ( X f c − X f ca )( V fR ) ∗ i ( V fR ) j . (4.10)Finally we switch to Dirac spinor notation for the fermions and introduce f a ≡ V / ( √ N ) tomatch to the standard normalization for the anomalous couplings. These are given by L anom = a ( x ) f a α s π G µν ˜ G µν + EN a ( x ) f a α em π F µν ˜ F µν , (4.11)with E/N = 8 / N DW = 2 N = 9). The couplings tofermions are given by L a = ∂ µ a f a f i γ µ (cid:104) C Vf i f j + C Af i f j γ (cid:105) f j , (4.12)with C Vf i f j = − g Lf i f j + g Rf j f i N = X f ca − X f a N δ ij + X f c − X f ca N ε fR,ij − X f − X f a N ε fL,ij , (4.13) C Af i f j = g Lf i f j + g Rf j f i N = X f ca + X f a N δ ij + X f c − X f ca N ε fR,ij + X f − X f a N ε fL,ij , (4.14)17nd the shorthand notation ε fL,ij ≡ ( V fL ) i ( V fL ) ∗ j , ε fR,ij ≡ ( V fR ) i ( V fR ) ∗ j . (4.15)Note that the diagonal elements of these parameters satisfy0 ≤ ε fL/R,ii ≤ , (cid:88) i ε fL/R,ii = 1 . (4.16)While the above expressions are valid for any axion model with PQ charges that are universalfor two fermion generations , in the present model these expressions simplify to C Vu i u j = ε uL,ij − ε uR,ij , C Au i u j = 2 δ ij − ε uL,ij − ε uR,ij , (4.17) C Vd i d j = ε dL,ij , C Ad i d j = 2 δ ij − ε dL,ij , (4.18) C Ve i e j = − ε eR,ij , C Ae i e j = 2 δ ij − ε eR,ij . (4.19)Using the approximate expressions in Eq. (2.10), the rotations have the parametric structure V uL ∼ V uR ∼ λ λ λ λ λ λ , V dL ∼ V eR ∼ λ λ λ λ λ λ , V dR ∼ V eL ∼ λ λ λ λ , (4.20)so that all relevant V i are CKM-like, and we have ε uL ∼ ε uR ∼ ε dL ∼ ε eR ∼ λ λ λ λ λ λ λ λ . (4.21)Therefore, the diagonal axial couplings are to very good approxmation independent of therotations, and we get, denoting C f i ≡ C Af i f i , C u = C d = C e = C c = C s = C µ = 29 , C t = 0 , C b = C τ = 19 . (4.22)The flavor-violating axion couplings are controlled by ε fij , whose numerical values, beyond theparametric suppression given above, are known for a given fit to masses and mixings. Besidesthere is an overall suppression factor 1 /f a that is proportional to the axion mass m a , withthe usual conversion factor for QCD axions as obtained from Chiral Perturbation Theory [26]and Lattice QCD [27] m a = 5 . µ eV (cid:18) GeV f a (cid:19) . (4.23) See Ref. [25] for a recent example where this structure is realized within a generalized DFSZ model, andcan be used to suppress the axion couplings to nucleons and electrons. .2 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 ofTable 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], lepton flavor-violating µ − e transitions contributing to µ → ea [30] and µ → eaγ decays [31, 32], (flavor-diagonal)axion-electron couplings bounded by the measurement of the WD luminosity function [33],and effective axion couplings to nucleons constrained from the burst duration of the SN 1987Aneutrino signal [34]. We did not include bounds from e.g. flavor-violating tau decays [35],since they give much weaker constraints.Coupling m max a /C [eV] m max , U(2) a [eV] f min , U(2) a [GeV] Constraint C µe . · −
78 7 . · µ → ea [30] C Vbs . · −
16 3 . · B + → K + a [28] C Vsd . · − . · K + → π + a [29] C Aee . · − . · WD Cooling [33] C N . · − . · SN1987A [34]
Table 13:
Bounds on selected axion-fermion couplings; here C µe ≡ (cid:113) ( C Vµe ) + ( C Aµe ) and C N ≡ (cid:113) C p + C n denotes the effective couplings to nucleons, with axion couplings to protonsand neutrons C p,n defined analogously to the axial vector couplings in Eq. (4.12). The secondcolumn denotes the model-independent upper bounds on the ratio of m a /C , where C denotesthe respective coupling, while the third and fourth columns contain the upper (lower) boundon m a ( f a ) in our model, using the numerical results for the couplings of Section 3.2, wherefor explicitness we took the fit QL ν M -1 (other fits give similar constraints). We have further used the predictions of the axion couplings in our model to obtain anupper bound on m a , or equivalently a lower bound on f a , which is shown in Table 13 forthe fit QL ν M -1 of the complete D × U (1) model in Table 10 (the result for the other fitsare very similar). As a result of the strong CKM protection of s − d transitions C Vsd ∼ λ ,the main constraint on the model comes from astrophysics, similar to flavor-universal axionmodels. Since the bound from WD cooling and SN1987A are comparable, and the precisevalue of the latter is debated in the literature (see e.g. the recent discussion in Ref. [36] whichfinds a constraint on m a /C roughly a factor 5 weaker than the PDG bound), we only takethe 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 > . · GeV. The predictions for thebranching ratio of K + → π + a decays are givenBR( K + → π + a ) = 4 . · − (cid:16) m a
14 meV (cid:17) , (4.24)which is far below the future sensitivity of NA62 [37, 38] given the constraint from WDcooling. This is in sharp constrast to the U(1) Axiflavon proposed in Ref. [11] (see alsoRef. [12]), where the d − s transition is only Cabibbo-suppressed, C Vsd ∼ 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 Matter19DM) abundance. One of the simplest scenarios is the misalignment mechanism [39–41], validwhen U (1) F is broken before inflation . At this stage the axion is essentially massless andtakes a generic field value misaligned from the vacuum value by an angle θ . Around the QCDphase transition the axion potential is generated, and the axion begins to oscillate aroundthe minimum. The energy density stored in these oscillations can be approximately relatedto the present DM abundance as [42]Ω DM h ≈ . (cid:18) µ eV m a (cid:19) . θ , (4.25)where θ ∈ [ − π, π ] is the initial misalignment angle. Thus for not too small values θ (cid:38) . π ,the natural window for axion DM is given by axion masses roughly between (1 ÷ µ eV,which correspond to axion decay constants f a ∼ (10 ÷ ) GeV and a cutoff in the range Λ ∼ (10 ÷ ) GeV. This range of axion masses preferred by DM through the misalignmentmechanism will be probed by the ADMX upgrade in the near future [43]. Indeed the discoveryprospects of the U (2) Axiflavon are mainly due to its coupling to photons, and we summarizethe status of the relevant experiments in the usual ( m a , g aγγ ) plane in Fig. 1, where g aγγ = | / − . | α em / (2 πf a ). Also cosmological scenarios with post-inflationary U (1) F breaking are viable, provided the presence of asuitable explicit breaking term to solve the domain wall problem arising from N DW = 9. This is in contrast tothe U(1) Axiflavon in Ref. [11], where the upper bound on the axion mass from K → πa prevents to obtainthe right amount of axion dark matter if U (1) F is broken after inflation. Repeating the numerical fit as in Section 3.2 with SM input values at 10 GeV, the χ and χ O (1) getslightly 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 overallpredictions change only marginally. - - - - - - - - - - - - - - - m a [ eV ] | g a γγ | [ G e V - ] �� ⇑ / ���� ��� ⇒ �������� + ���� - �� ���������� ���� - �� � ( � ) �������� Figure 1:
Prediction of the axion-photon coupling as a function of the axion mass m a .The yellow band denotes the usual axion band of KSVZ models with a single pair of vector-like fermions, taken from Ref. [44]. The red line denotes the parameter space of the U(2)Axiflavon model, which extend up to the • mark, denoting the bound from WD cooling, seeTable 13. Also shown are the bounds from structure formation excluding hot DM (HDM) [45–47], the bound from the evolution of Horizontal Branch (HB) stars in globular clusters [48],the expected sensitivity of the ALPS-II experiment [49], the present and future bounds fromAxion helioscopes provided by CAST [50] and IAXO [51, 52], and from Axion Haloscopes likeADMX [53, 54], MADMAX [55] and the planned ADMX upgrade [43]. In summary, we have a proposed a U (2) F model of flavor with horizontal quantum numberscompatible with an SU (5) GUT structure. The flavor symmetry U (2) F loc . (cid:39) SU (2) F × U (1) F isspontaneously broken by two flavon fields φ and χ , which transform as a doublet and singletunder SU (2) F , respectively. Similarly, the three generations of SM fermions transform as + of SU (2) F , and there is a simple assignment of U (1) F quantum numbers X = 0 , X a = X a = X = − X φ = − X χ = 1 . (5.26)The SM Yukawas arise from higher-dimensional operators made invariant under U (2) F byappropriate insertions of flavons, suppressed by the cut-off scale Λ (cid:29) v . In this way thehierarchical structure of Yukawa matrices is explained by powers of two small parametersthat control the breaking of U (2) F , 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 structurewith three texture zeros in the 1-1,1-3 and 3-1 entries, while the 1-2 entry is antisymmetric, seeEq. (2.1). The presence of these textures leads to accurate relations between CKM elementsand 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 down21uark sector. Indeed we have obtained a very good fit to fermion masses and mixings withcoefficients that are O (1) (all between 0.4 and 2), see Table 4.We have then included the neutrino sector, which gives a consistent fit to experimen-tal data only with Dirac neutrinos. To this extent, we have introduced three right-handedneutrinos (SM singlets), which also transform as + of SU (2) F and have equal chargesunder U (1) F . The resulting structure of the Dirac mass matrix (cf. Eq. (2.17)) has againthree texture zeros and only weak inter-generational hierarchies, thus predicting large mixingangles. The U (1) F charge of the singlets enters only in the overall suppression factor andcan account for the smallness of neutrino Yukawas if taken to be 5 ÷
6. The combined fit tothe complete fermion sector is viable only for neutrinos with normal mass hierarchy, and stillshows a good performance with O (1) coefficients between roughly 1 / ÷ β -decays is far below futureexperimental sensitivities.In order to have a consistent scenario with Majorana neutrinos, we have futhermore dis-cussed an D × U (1) variant of the U (2) F model, where the SU (2) F factor is replaced by adiscrete D 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 sectorare exactly the same as in the U (2) F case, up to a sign flip in the 1-2 entry that is largelyirrelevant. This sign flip however allows for an unsuppressed 1-2 entry in the Weinberg op-erator, 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 neutrinomass matrix, so that the SU (5) structure connects large leptonic mixing angles to small mix-ing angles in the quark sector. Indeed, the parametric flavor suppression of up-, down-quark,charged lepton and neutrino masses follows the simple pattern m { u,d,e,ν } ∼ ε ε ε { ε, ε , ε, ε } { ε, ε, ε , ε } { , ε, ε, ε } , (5.27)where the mass scale is set by v in the quark and charged lepton sector and v /M in theneutrino sector. The difference between the fermion sectors just follows from the different U (1) charge assignments for the third generation, see Eq. (5.26). Although this model is morepredictive than the Dirac case, since two U (1) charges are replaced by a single mass scale M , we obtain an excellent fit 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 inthe previous case only neutrinos with normal mass hierarchy are viable. Scanning over manygood fits, we have obtained a slightly narrower range for the sum of neutrino masses roughlygiven by (58 ÷
78) meV, while again the predictions for the effective neutrino mass enteringbeta 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 thepredictions in the neutrino sector. In general, sizable deviations in experimental observables22rom the SM require the existence of sufficiently light degrees of freedom. While there isno particular reason why the cutoff and its associated dynamics should be light, there isthe natural possibility to solve the strong CP problem and account for DM through theGoldstone boson of the global U (1) F symmetry, which we refer to as the U (2) Axiflavon. Incontrast to the Axiflavon from a single Froggatt-Nielsen U (1) F symmetry [56] as presented inRefs. [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/KSVZaxion, 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 explainedthrough the misalignment mechanism, implying axion masses around (1 ÷ µ eV, whichcorresponds to a cutoff scale of roughly (10 ÷ ) GeV. This range will be tested by futureaxion haloscope searches.The present model could be extended in several ways: 1) A more careful study of theneutrino sector might allow to pin down the predictions analytically, and it could be interestingto 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 hierarchyproblem, possibly in connection with a full SU (5) GUT, trying to relate GUT breaking scale,flavor breaking scale and the axion decay constant, similar to Ref. [57]. 3) Finally, it might beinteresting to study possible UV completions and calculate the low-energy constraints fromflavor-violating obervables on the new dynamics. Acknowledgements
We thank F. Feruglio, J. Lopez Pavon, A. Ringwald, A. Romanino and A. Trautner for usefuldiscussions and comments. ML acknowledges the support by the DFG-funded Doctoral School”Karlsruhe School of Elementary and Astroparticle Physics: Science and Technology”.23 D and D Group Theory
In this Appendix we provide some details about the structure of the dihedral groups D and D and fix the notation for constructing group invariants (see also Refs. [5, 58, 59]).The dihedral group D is the symmetry group of an equilateral triangle and is isomorphicto S , the permutation group of three objects with order 6. The group is generated by twoelements R and S , where R is the rotation through 120 ◦ and S is the reflection about one ofthe bisectors. Since R = S = 1 and SR = R S , the six elements are 1 , R, R , S, RS, SR . D has two one-dimensional representations , (cid:48) and one two-dimensional representation .The representation matrices for R and S can be chosen as in Table 14. The tensor productsRepresentation R S (cid:48) − (cid:32) e πi e − πi (cid:33) (cid:18) (cid:19) Table 14:
Representation matrices for D . of two one-dimensional representations decompose as follows: ⊗ = , ⊗ (cid:48) = (cid:48) , (cid:48) ⊗ (cid:48) = , (A.1)while for the product of two ’s one gets ⊗ = ⊕ (cid:48) ⊕ . (A.2)For two doublets ψ = (cid:18) ψ ψ (cid:19) and ϕ = (cid:18) ϕ ϕ (cid:19) one finds( ψ ⊗ ϕ ) = ψ ϕ + ψ ϕ , ( ψ ⊗ ϕ ) (cid:48) = ψ ϕ − ψ ϕ , ( ψ ⊗ ϕ ) = (cid:18) ψ ϕ ψ ϕ (cid:19) . (A.3)In the following we will use the simplified notation for singlet components (i.e. invariants)( ψ · ϕ ) ≡ ( ψ ⊗ ϕ ) = ψ ϕ + ψ ϕ . (A.4)From a given doublet ϕ one can construct another doublet (cid:101) ϕ = σ ϕ ∗ = (cid:18) ϕ ∗ ϕ ∗ (cid:19) , with invariant( (cid:101) ϕ · ϕ ) = ϕ ∗ ϕ + ϕ ∗ ϕ . (A.5)Note that because of Eq. (A.2) any product of doublets contain at least one singlet. For threedoublets it is given by ( ψ · ϕ · χ ) = ψ ϕ χ + ψ ϕ χ , (A.6)while there are three different singlets in the product of four doublets, which we define as( ψ ⊗ ϕ ⊗ χ ⊗ η ) = ψ ϕ χ η + ψ ϕ χ η ψ ϕ χ η + ψ ϕ χ η ψ ϕ χ η + ψ ϕ χ η . (A.7)24or the case of ψ = ϕ and χ = η there are just two invariants for which we use the notation:( ψ ⊗ ψ ⊗ χ ⊗ χ ) = (cid:40) ( ψ · ψ )( χ · χ ) ≡ ψ ψ χ χ ( ψ · ψ · χ · χ ) ≡ ψ χ + ψ χ . (A.8)Finally we turn to the dihedral group D which is the symmetry group of regular hexagon.It is isomorphic to D × Z , and therefore inherits the group theoretical structure discussedabove, except that each representation carries an additional Z charge, which is conservedin tensor decompositions. Thus, we have four one-dimensional representations + , − , (cid:48) + , (cid:48)− (where + denotes the total singlet) and two two-dimensional representations + , − . Thedecompositions of these representations follow from the D ones, for example we have − ⊗ − = + ⊕ (cid:48) + ⊕ + , + ⊗ − = − ⊕ (cid:48)− ⊕ − . (A.9)Therefore in D the tensor product ( − ⊗ − ⊗ − ) does not contain a singlet. B Fit Results
Parameter QL ν D -1 QL ν D -2 QL ν D -3 QL ν D -4 QL ν M -1 QL ν M -2 λ u λ u λ u λ u -1.103 -1.419 1.511 2.422 1.196 1.615 λ u δ -0.640 -0.720 -3.948 -1.097 -3.837 -3.988 λ d λ d -1.000 -1.156 -1.075 -0.972 -0.973 0.976 λ d λ d -0.355 0.401 0.414 0.423 0.365 -0.394 λ d λ (cid:96) λ (cid:96) λ (cid:96) λ (cid:96) -0.992 -1.132 1.175 1.193 -1.198 1.294 λ (cid:96) λ ν λ ν -0.994 -1.303 0.325 0.398 -0.844 -0.760 λ ν -2.588 -1.074 -1.505 1.681 1.137 -1.078 λ ν (cid:113) (cid:113) λ ν X N a X N3 v/M × -0.421 -0.520 ε φ ε χ Table 15:
Fit parameters for Dirac ( SU (2) × U (1) Model) and Majorana neutrinos ( D × U (1)Model). The parameters are defined in Eqs. (2.1) and (2.17), and Eq. (3.8), respectively. O i = { y u , y d , . . . } we define the tuning ∆ i and the pull P i as∆ i = max j (cid:12)(cid:12)(cid:12)(cid:12) ∂ log O i ∂ log p j (cid:12)(cid:12)(cid:12)(cid:12) , P i = O fit i − O exp i σ exp i , (B.1)where p j = { λ u,d,(cid:96),νij , ε φ , ε χ , M } are the fit parameters. For the sake of brevity, we restrict toFit 3 and 4 in the Dirac case, the other two fits give similar results. As can be seen fromTables 16 and 17, the tuning of the observables is quite low, at most 10% for the Dirac caseand about 20% in the Majorana case. As expected from the χ value, the pulls are small andare dominated by the quark Yukawas (and in the Majorana case also by the PMNS mixingangles). Observable ∆(QL ν D -3) Pull(QL ν D -3) ∆(QL ν D -4) Pull(QL ν D -4) y u y c y t y d y s y b y e y µ y τ θ CKM13 θ CKM12 θ CKM23 δ CP m m θ PMNS13 θ PMNS12 θ PMNS23
Table 16:
Fine-tuning and pulls for the observables of fit QL ν D -3 and fit QL ν D -4. bservable ∆(QL ν M -1) Pull(QL ν M -1) ∆(QL ν M -2) Pull(QL ν M -2) y u y c y t y d y s y b y e y µ y τ θ CKM13 θ CKM12 θ CKM23 δ CP m m θ PMNS13 θ PMNS12 θ PMNS23
Table 17:
Fine-tuning and pulls for the observables of fit QL ν M -1 and fit QL ν M -2. C Scalar Potential
In this section we consider an explicit scalar potential that generates the VEVs we haveassumed in Section 3, serving merely as a proof of existence. In particular, this potentialshould be reassessed in a UV complete setup, possibly in connection with a supersymmetric SU (5) GUT.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 continuous globalsymmetry that can be identified with U (1) F . The transformation properties under D [ U (1) F ]are φ = 2 − [ − , χ = 1 + [ − , ψ = 1 − [+1] , (C.1)and the most general, renormalizable scalar potential for these fields is given by V scal = m χ | χ | + (cid:0) m φ + κ χ | χ | + κ ψ | ψ | (cid:1) ( ˜ φ · φ ) + m ψ | ψ | + λ φ · ˜ φ )( φ · φ ) + λ φ · ˜ φ · φ · φ ) + λ | χ || ψ | + λ χ | χ | + λ ψ | ψ | + (cid:20) κ ψψ ( φ · φ ) + κ χ ∗ χ ∗ ( φ · φ ) + 12 λ χψ ψψχχ + ρψ ( ˜ φ · φ · φ ) + h . c . (cid:21) , (C.2)where the D singlet contractions are explained in Appendix A and we take κ , κ , λ χψ and ρ to be real. The ground state of this potential is most easily studied in the limit when ρ (cid:28) , κ (cid:28) , λ χψ (cid:28) . (C.3) We do not include the SM Higgs, because its backreaction on the flavon potential is negligible as the flavonVEVs are much larger than the electroweak scale. In turn, the flavons will generate a large mass term for theHiggs, which is just the usual hierarchy problem that we do not address here. ρ and κ is given by v = λ χ m φ − κ χ m χ κ χ − λ λ χ , v χ = λ m χ − κ χ m φ κ χ − λ λ χ . (C.4)There is a symmetry exchanging φ ↔ φ in the potential, which are connected by a D transformation that we can use to assume the large VEV in the φ direction without loss ofgenerality. The VEVs of v and v ψ only arise at O ( κ ) and O ( κρ ), respectively: v = κ v χ ( λ − λ ) v χ v , v ψ = κ ρ v ( λ − λ ) (cid:32) v χ ( κ χ − λ λ χ )˜ m (cid:33) , (C.5)with the shorthand notation˜ m = κ χ ( κ χ m ψ − κ ψ m χ − λ m φ ) + κ ψ λ χ m φ + λ ( λ m χ − λ χ m ψ ) . (C.6)In order to suppress the VEVs of φ and ψ sufficiently, i.e. to ensure the validity of e.g.Eq. (2.1), we need roughly v /v ∼ v /v χ ∼ κ (cid:46) ε χ ∼ − . Such a small coupling istechnically natural, since in the limit of κ → , λ χψ → ρ →
0) the Lagrangian acquiresa larger symmetry. This can be seen from spelling out the third line of the potential explicitly: V scal ⊃ κ ψ φ φ + κ χ ∗ χ ∗ φ φ + λ χψ χ ψ + ρψ (cid:0) φ φ ∗ + φ φ ∗ (cid:1) + h . c . (C.7)Indeed, this part only breaks the additional U (1) symmetry of the scalar kinetic terms(besides the remaining U (1) F ) if ρ (cid:54) = 0 and κ (cid:54) = 0 or λ χψ (cid:54) = 0.This observation is also crucial to understand why the additional field ψ is needed: itscoupling ρ is the only parameter that breaks the U (1) symmetry under which χ is neutraland φ and φ carry opposite charges. Moreover, it makes clear that we expect (in additionto the massless U (1) F Goldstone) a very light pseudoscalar in the spectrum whose mass issuppressed by the small couplings κ , λ χψ and ρ .After these analytical considerations we finally provide a numerical example, taking thefollowing set of parameters: m φ = − m , m χ = − / m , m ψ = 2 m , λ = 1 , λ = 1 / , λ χ = 1 , κ χ = − / λ = 2 / , λ χψ = − / , κ = − / , κ ψ = 7 / , ρ = − / , λ ψ = 9 / , κ = 1 / v = 4 . m , v = − . · − m , v χ = 1 . m , v ψ = − . · − m , (C.9)and therefore ε φ = 0 . (cid:18) m Λ (cid:19) , ε χ = 0 . (cid:18) m Λ (cid:19) . (C.10)Finally, the scalar mass spectrum is given by one massless Goldstone, 6 massive scalars withmasses { . , . , . , . , . , . } m and a light scalar with mass 2 . · − m . Using the lowerbound on Λ from Section 4.2 (corresponding to an axion mass m a (cid:46)
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