Predictions of m ee and neutrino mass from a consistent Froggatt-Nielsen model

The seesaw mechanism is the most attractive mechanism to explain the small neutrino masses, which predicts the neutrinoless double beta decay (0 νββ ) of the nucleus. Thus the discovery of 0 νββ is extremely important for future particle physics. However, the present data on the neutrino oscillation is not sufficient to predict the value of m ee as well as the neutrino mass m iν . In this short article, by adopting a simple and consistent Froggatt-Nielsen model, which can well explain the observed masses and mixing angles of quark and lepton sectors, we calculate the distribution of m ee and m iν . Interestingly, a relatively large part of the preferred parameter space can be detected in the near future.


I. INTRODUCTION
The Standard Model of particle physics contains 28 free parameters, including the neutrino sector, which could not be explained theoretically.Among them, Yukawa couplings have a hierarchy structure, that is the flavor puzzle, which has attracted theorists' attention for decades [1][2][3][4][5][6][7][8].Meanwhile, for the neutrino sector, the neutrino oscillation experiments, e.g.Super-K [9], SNO [10], and Daya Bay [11], have shown that neutrinos are not massless but possess tiny mass m i ν (i = 1, 2, 3).Among different explanations, the seesaw mechanism [12][13][14][15] is regarded as the most natural and promising one.An important corollary to the seesaw mechanism is the neutrinoless double beta decay (0νββ), which is closely related to the effective Majorana mass m ee .Therefore, the discovery of 0νββ will be a huge breakthrough for the particle physics community.Nevertheless, neither the seesaw mechanism nor neutrino oscillation experiments can tell the values of m ee and m i ν .This could be viewed as another intriguing puzzle.In this work, we try to propose a simple and consistent model that can well explain the fermion mass hierarchy and predict the values of m ee and m i ν simultaneously.It is well known that the Forggatt-Nielsen (FN) mechanism [16][17][18] provides an excellent method to explain the flavor puzzle, in which the Standard Model (SM) gauge group is extended by a horizontal global U (1) FN symmetry.The U (1) FN is broken by a vacuum expectation value (VEV) of a new scalar field ϕ whose U (1) FN charge is −1.Naturally, a dimensionless parameter λ can be defined, i.e. λ = ⟨ϕ⟩/M PL ∼ O(0.1), where M PL ≃ 2.4×10 18 GeV is the reduced Planck scale.All SM particles also carry the U (1) FN charge, and the value of the FN charge is generation dependent, which indicates that the masses of different generations of particles get suppressed by different powers of λ.Thus, the hierarchy issue could be well explained by the FN mechanism (see Ref. [8] for a very recent review).
Obviously, the core of the FN mechanism is the assignment of FN charge for SM particles.Recently, there are some works that did a blanket search to find optimal FN charge assignment [19,20].Especially, in Ref. [19] the advanced reinforcement learning technics are involved.In our work, instead of adopting these kinds of brute force methods, we attempt to fix the FN charge of SM particles by doing a qualitative analysis.The rationality of the FN charge assignment is evaluated by comparing the theory predictions with experimental observations.In fact, it turns out that our strategy is quite effective, and to some extent, our results are in good agreement with previous blanket scan results [20].
As for the more interesting neutrino sector, the seesaw mechanism actually implies that the neutrino masses can be produced from a dimension-five effective operator [14,21,22], which can be derived by integrating out the heavy right-handed neutrino states.In this case, the neutrino mass and mixing angle are also affected by the FN charge, since ν i L belongs to the electroweak doublet ℓ i L and also carries FN charges.Therefore, we show that it is possible to handle the flavor and neutrino puzzle within a unified FN framework.Interestingly, combined with the measurements of neutrino mass square difference ∆m 2  21 , |∆m 2 32 | [23], and some cosmological constraints on m i ν [24], we can calculate the distribution of m ee and m i ν .Surprisingly, we find that our predictions on m i ν are quite consistent with the available experimental data.Besides, a relatively large parameter space of m ee of our model could be explored in the near future neutrinoless ββ decay experiment, e.g.LEGEND-1000 [25].
This paper is organized as follows.In Sec.II we give a brief introduction to the FN mechanism and the analysis of how to fix the FN charges.In Sec.III we calculate the predictions of our model on m ee and m i ν as well as the near future constraints from LEGEND-1000.Conclusions and further discussions are given in Sec.IV.

II. THE CONSISTENT FORGGATT-NIELSEN MODEL
The FN model we are considering is a simple extension of the Standard Model [16][17][18].The mass matrices of quarks are granted by Yukawa couplings, which are where H is the Standard Model Higgs doublet and H = iσ 2 H * .Under our FN framework, the Yukawa couplings can be expressed as where the N is an overall factor to accommodate the overall scale difference between up-type and down-type sectors, whose origin could be two-Higgs-doublet models at high energy [26], the g ij is the universal coupling, whose magnitude |g ij | fulfills a normal distribution N (µ, σ 2 ), while its argument fulfills a uniform distribution from 0 to 2π.In our work, we choose µ = 1 and σ = 0.3 for a benchmark case.Since we have set U (1) H FN = 0 and U (1) ϕ FN = −1, the value of n ij u/d is determined by the FN charge of quarks, Clearly, once we fix the value of λ and quarks' FN charge, the quark mass, mixing angle, and CP angle are almost fixed.Mass hierarchy is indicated by the fermion mass ratio between generations, i.e. m u /m t , m d /m b , and so on.We find that if we focus on the mass ratios, then the charge assignment will become much easier.Take Q i L and d j R for example, where the most general form of their FN charge should be U (1) The overall factor λ c+f will not affect the fermion mass ratio and mixings.However, this factor could also be used to explain the absolute quark mass (such as N in Eq. ( 1)), i.e. the mass hierarchy between up-type and down-type quarks.In the following content, we just set the FN charge of third-generation fermion equal to zero, which is equivalent to absorbing the λ c+f factor into N , and we will comment on this issue in Sec.IV.For simplicity, we only consider the FN charge to be an integer or half-integer less than 5. Similar conditions are also adopted in previous literature [19,20].For the quark sector, one could show that the Cabibbo-Kobayashi-Maskawa (CKM) matrix is mainly determined to produce such a mixing pattern [20,28].Once we know the U (1) FN , the FN charge of u R and d R can be roughly fixed by comparing with the observed quark mass ratios.
In Table I we have summarized all available mass ratios, mixing angles, and CP angles of quark and lepton sectors.Considering that the U (1) FN was broken at a very high energy scale ∼ M PL , all the numbers in Table I should also be evaluated at a high energy scale.From Refs.[29,30], we can see that the mass ratio of the quark and lepton sectors are almost energy independent as long as the energy scale larger than ∼ 10 8 GeV.Therefore, we can safely substitute the mass ratio at ∼ 10 12 GeV for the results at ∼ M PL .As for the mixing angles and CP angle, we assume that they are energy independent.
Assuming the FN charges of u R and d R are {a, b, 0} and {c, d, 0}, respectively, we can derive that From Table I we roughly have where the λ ′ ∼ 0.2.Comparing Eq. ( 5) and Eq. ( 6), we can derive that For the lepton sector, the mass terms are generated by Yukawa couplings and a five-dimensional operator, that is where Note that here we use the same N as Eq. ( 2) because of the fact that m b ∼ m τ at a very high energy scale [29].Besides, the g ′ ij is a symmetric matrix due to the Majorana nature of the neutrino.Similar to the quark sector, the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix is mainly determined by ℓ i L .The observations tell us that the mixing angles θ P 12 and θ P 23 are relatively larger compared to θ P 13 , and θ P 12 is slightly smaller than θ P 23 .Following a similar logic and using the universal λ ′ , we assign the FN charge for the lepton doublet as U (1) ℓL FN = {1, 0.5, 0}.One interesting fact is that we can easily prove that the rank of n ν is 1, which means there would be two mass eigenvalues being almost zero and one relatively large eigenvalue after diagonalization.This indicates that the FN mechanism naturally prefers normal order (NO) 1 .Therefore, in the following content, we just stick to the NO scenario.
Then U (1) eR FN is obtained from estimating the charged lepton mass ratios.Specifically, assuming U (1) eR FN = {e, f, 0} we have From Table .I we roughly have where the λ ′ ∼ 0.2.Comparing Eq. ( 10) and Eq. ( 11), we can derive that Until now, by doing a qualitative analysis we have fixed the FN charge of SM particles (see Table II).However, there are two issues that need to be emphasized.The 1 Another perspective to understand this feature is through the FN charge assignment of ℓ L .Combining Eq. ( 8) and U (1) ℓ L FN = {1, 0.5, 0} (see Table II), the ratio of three eigenvalues of y ij ν is roughly 1 : first one is the global U (1) FN symmetry.The above analysis is based on an assumption that all terms in Eq. ( 1) and Eq. ( 8) respect the global U (1) FN .However, it is believed that any global symmetries must be broken by nonperturbation effects in quantum gravity [32].With the charge assignment in Table II, we found there is a discrete Z 33 symmetry, actually this symmetry is anomaly-free for Z 33 × [SU (2) L ] 2 and Z 33 × [SU (3) c ] 2 , and therefore can be gauged.This can be regarded as quite an interesting feature of our model.
Another issue is the exact value of λ, which is the only free parameter after fixing the FN charge.We conduct the analysis by using a very rough number, i.e. λ ∼ λ ′ ∼ 0.2, while a more accurate λ is necessary for a concrete FN model.In the following content, we adopt the minimum chi-square method to find the best value of λ.
The strategy is quite straightforward.As we mentioned above, in our model the g ij is the universal coupling whose magnitude fulfills a normal distribution N (1, 0.3), while its argument fulfills a uniform distribution from 0 to 2π.With the fixed FN charges and λ, all the couplings, e.g., y ij u , y ij d , y ij ℓ , y ij ν , can be generated.Then, all the desired quantities (denoted X i ), including the quark and lepton mass ratios, the mixing angles, and CP angles, will be fixed.By randomly generating g ij we can derive the distribution X i .The chi square is defined as where X exp i is the experimentally measured value (see Table I), E(X i ) is the expectation value of X i , while V (X i ) is the deviation.Here we take where δ P is not the direct observable.By scanning the parameter space of λ we can find the best value that minimizes the χ 2 (λ).
To calculate Eq. ( 13) we need the exact distribution of X i .For the quark sector, the Yukawa matrices can be decomposed as where U u,d and W u,d are unitary matrices, D u,d is a diagonal matrix with all real elements.The U u and W u are obtained from  I.
Within our notation, the CKM matrix can be expressed as which contains all information on quark mixing angles (θ C ij ) and CP angle (δ C ).Under standard parametriza-tion, the U CKM is where V SP is a unitary matrix possessing four real parameters, that is where c ij = cos θ ij and s ij = sin θ ij .Note that we adopt the convention that θ ij ∈ [0, π/2) and δ ∈ [−π, π).Utilizing Eq. ( 15) and Eq. ( 18), we can calculate quark sector parameters.One significant difference for the lepton sector (see Eq. ( 8)) is that the neutrino mass is generated by a fivedimensional effective operator.As we mentioned in this case y ν is a complex symmetric matrix.Similar to the quark sector, we do the following decomposition, i.e., where D ℓ and D ν are diagonal matrices with all real elements and U ℓ , W ℓ , and U ν are unitary matrices.The U ℓ and W ℓ are obtained from Then the mass ratios for charged leptons are Next, we need to derive the explicit form of U ν .Define Ũν such that y † ν y ν = Ũν (D ν ) 2 Ũ † ν , and since y ν is a symmetric matrix, we can derive that y ν = Ũν ΦD ν Ũ T ν , where Φ is a diagonal matrix and each element is a pure phase.Compared with Eq. ( 19) we have U ν = Ũν Φ −1/2 .In our notation, the PMNS matrix can be written as Different from the CKM matrix, one could only rotate three phases from charged leptons, which results in two  I.
extra phases in U PMNS compared to the CKM matrix, where α M and β M are the Majorana phases.In our work, we only consider θ P ij and δ P .So now we know how to calculate X i , and then by repeating random sampling g ij and g ′ ij , we can get the distribution of X i .Combining with Eq. ( 13) we do a parameter scan to get the best value of λ, that is λ = 0.171 with χ 2 = 4.69 .
Under such an input, the probability density function (PDF) of the quark mass ratios are plotted in Fig. 1, and the results of mixing angles and CP angle are shown in Fig. 2. The black dashed lines indicate the experimental measurements (see Table I).It shows that our model predictions all agree with the experimental observations.Similarly, Fig. 3 and Fig. 4 show the PDF of the lepton mass ratio and mixing angles.For neutrinos, there are only mass square differences available, i.e. ∆m 2  21 = (m 2 ν ) 2 − (m 1 ν ) 2 and ∆m 2 32 = (m 3 ν ) 2 − (m 2 ν ) 2 .Note that δ P is not a directly observable quantity and has a large uncertainty, so it is not included in χ 2 (λ).In conclusion, adopting the FN charge in Table II and λ = 0.171, our FN model could successfully explain 15 parameters in the standard model.In comparison, we also calculate the χ 2 by using the best three charge assignments in Table.I of Ref. [20], and we find that our model has a smaller χ 2 .24), would be violated.

III. PREDICTIONS ON m i ν AND mee
In Sec.II, we have built a consistent FN model, and its predictions of 15 parameters agree very well with experimental observations.Based on this success, we are more interested in its predictions on the neutrino sector, especially, m i ν and m ee .As we mentioned in Sec.II, the FN mechanism naturally prefers the NO.As a cross-check, we randomly generate a neutrino Yukawa matrix, y ν , whose eigenvalues are {D i ν }.If the average of {D 1 ν , D 2 ν , D 3 ν } is smaller than the median, y ν is inverted order (IO).By sampling 10 6 times using our FN charge with λ = 0.171 and σ = 0.3, we find that NO takes up ∼ 98%.Therefore, we only consider NO in this work.and effective Majorana mass element mee.Note that m3 = 0.05 eV is adopted due to the cosmology and oscillation experiments constraints (see Fig. 5).
Based on the success of our FN model, we also explore its prediction on the neutrino sector, especially, the value of m i ν and m ee .We find that the FN mechanism naturally prefers the NO scenario, which is determined by its mathematical structure.Utilizing the neutrino oscillation and cosmology constraint, we calculate the mass spectrum in Fig. 5.It shows that m 3 ν is almost a constant, i.e. m 3 ν ∼ 50 meV.By adopting this benchmark value we predict that m 1 ν and m 2 ν are ∼ 1.6 meV and ∼ 9 meV respectively, which is perfectly consistent with the current observations.In addition, our model also gives a relatively precise constraint on m ee , i.e. 0.503 meV ≲ m ee ≲ 18.0 meV at 95% C.L.More interestingly, our model can be explored in a near future 0νββ experiment, such as LEGEND-1000 [25].
In Sec.II we introduce the factor N to handle mass hierarchy between up-type quark and down-type quark and the charged lepton (see Eq. ( 2) and Eq. ( 9)).One possible origin of N is that there are two different Higgs, like two-Higgs-doublet model [26] and supersymmetric theories at high energy.In fact, the FN charge could play the same role as N , e.g. the λ c+f in Eq. (4).For example, we find that if we reset U (1) d R FN = {3.5, 2.5, 2.5}, U (1) e R FN = {6, 3, 2}, then N is not necessary.However, the cost of this is that the discrete gauge symmetry Z 33 disappears.
The most distinguished feature of this work is the universality of the coupling g ij (including the g ′ ij for the neutrino sector).By combining this universality with the FN mechanism, we can successfully explain 15 SM parameters.Furthermore, its prediction on neutrino mass m i ν is surprisingly consistent with the current observation.Although the deep meaning behind this universality is unknown, we expect that there may be some hidden principles, and we save further explorations for future works.

FIG. 5 .
FIG.5.Neutrino mass spectrum under the constraint from cosmology and oscillation measurements.The gray vertical dotted line indicates the upper bound on m1, above which the cosmological bound, i.e.Eq.(24), would be violated.

TABLE II .
FN charge of quarks and leptons.