Probing Models of Neutrino Masses via the Flavor Structure of the Mass Matrix

We discuss what kinds of combinations of Yukawa interactions can generate the Majorana neutrino mass matrix. We concentrate on the flavor structure of the neutrino mass matrix because it does not depend on details of the models except for Yukawa interactions while determination of the overall scale of the mass matrix requires to specify also the scalar potential and masses of new particles. Thus, models to generate Majorana neutrino mass matrix can be efficiently classified according to the combination of Yukawa interactions. We first investigate the case where Yukawa interactions with only leptons are utilized. Next, we consider the case with Yukawa interactions between leptons and gauge singlet fermions, which have the odd parity under the unbroken Z_2 symmetry. We show that combinations of Yukawa interactions for these cases can be classified into only three groups. Our classification would be useful for the efficient discrimination of models via experimental tests for not each model but just three groups of models.


I. INTRODUCTION
Thanks to the discovery of a Higgs boson h at the CERN Large Hadron Collider (LHC) [1], we have entered the era to explore the origin of particle masses. Coupling constants of W ± , Z, t, b, and τ with h are measured at the LHC [2], and they are consistent with predicted values in the Standard Model (SM). These results strongly suggest that masses of gauge bosons and charged fermions are generated by the vacuum expectation value of the Higgs field, which provides h, as predicted in the SM. Thus, the mechanism to generate their masses in the SM was confirmed. On the other hand, neutrino masses are not included in the SM although neutrino oscillation data uncovered that neutrinos have their masses [3,4].
It is easy to add neutrino mass terms m ν ν L ν R to the SM similarly to the other fermion mass terms by introducing right-handed neutrinos ν R . However, since the neutrino is a neutral fermion in contrast to the other fermions in the SM, another possibility of its mass term exists. That is the Majorana mass term, (1/2)m ν ν L (ν L ) c . This unique possibility could be the reason why neutrinos are much lighter than the other fermions. New physics models for the Majorana neutrino mass can be found in e.g. Refs. .
The overall scale of the neutrino mass matrix m ν generated in new physics models is determined by the structure (tree level, one-loop level, and so on) of the diagram to generate m ν , masses of new particles in the diagram and coupling constants in the diagram. This means that the determination of the overall scale of m ν requires to specify many parts of the Lagrangian of each model. On the other hand, the flavor structure (ratios of elements) of m ν is simply determined by the product of Yukawa coupling matrices and fermion masses.
Thus, models to generate m ν can efficiently be classified according to the combination of Yukawa coupling matrices and fermion masses without the detail of these models. When we construct a new model to generate neutrino masses, it will be noticed indeed that the flavor structure is the key to find an appropriate set of model parameters although the overall scale of m ν can be easily tuned by using some parameters in the scalar potential.
In this letter, we first classify models for Majorana neutrino masses according to combination of Yukawa interaction between leptons without introducing new fermions. Next, we do the classification for the case where gauge singlet fermions are introduced such that they have the odd parity under the unbroken Z 2 symmetry which can be utilized to stabilize the dark matter. For Yukawa interactions of these new fermions with leptons, Z 2 -odd scalars are also introduced. We find that models can be classified into only three groups. The classification could be useful to approach efficiently the origin of Majorana neutrino masses with experimental tests of not each model but each group of models.
Models of neutrino masses can also be classified according to topologies of diagrams [58] or decompositions of higher mass-dimensional operators [59]. They seem useful to find new models and increase the number of models in order to exhaust all possibilities. In contrast with these classifications, ours would be useful to simplify the situation where many models exist.

II. CLASSIFICATION OF FLAVOR STRUCTURE
First, we introduce only scalar fields listed in Table I, which have Yukawa interactions with leptons. We do not always introduce all of them, and we utilize only scalar bosons for required Yukawa interactions. For the Yukawa interaction with the second SU(2) L -doublet scalar field Φ 2 , the flavor changing neutral current is forbidden by utilizing a softly-broken Z 2 symmetry as usually done in the two Higgs doublet models. In order to obtain m ν , we try to connect ν L to (ν L ) c by using these Yukawa interactions and the weak interaction. We do not care how scalar lines are closed because we concentrate on the flavor structure of m ν .  Each charged lepton (ℓ L , ℓ R , (ℓ L ) c , (ℓ R ) c ) should appear only once on the fermion line in order to obtain the simplest combinations, which would give the largest contribution to m ν .

Ä´
In addition, ℓ L and ℓ R should appear only in the next to each other on the fermion line. If they do not, the replacement of the structure between them with the mass term of ℓ can give the simpler combination 1 . It is assumed that m ν is generated via a solo mechanism (a solo kind of fermion lines). Then, we find that only the following five combinations 2 connect where Yukawa matrices Y A , Y s S , y ℓ , and Y ∆ S are defined in Table I. Diagrams of fermion lines for combinations in eqs. (1)-(5) are shown in Figs. 1-5, respectively. The SU(2) L gauge coupling constant g 2 is shown for clarity although the weak interaction is flavor blind. The combination in eq. (3) gives at least a dimension-9 operator for the Majorana neutrino mass 1 Although the electron Yukawa coupling is small, the diagonal matrix y ℓ would not be negligible because of the tau Yukawa coupling. 2 Notice that another possible combination Y s  while the others can be a dimension-5 one.
The combination in eq. (5) is the one in the Zee-Wolfenstein model [5,6] of the Majorana neutrino mass at the one-loop level, which has been excluded already by the neutrino oscillation data [60]. Thus, this combination is ignored below. An example for m ν in eq. (1) is the Zee-Babu (ZB) model [7,8], which generates m ν at the two-loop level. The structure in eq. (2) is given in a model in Ref. [9] by Cheng and Li (the CL model), which also generates m ν at the two-loop level 3 . The Gustafsson-No-Rivera (GNR) model [10] is an example for the combination in eq. (3), in which m ν is generated at the tree-loop level. Scalar lines of W + and s −− are connected at the one-loop level by introducing the unbroken Z 2 symmetry and Z 2 -odd scalar fields, which provide a dark matter candidate. The structure in eq. (4) is given at the tree level, and an example is the Higgs triplet model (HTM) [9,11]. Since eqs. (2) and (3) have the same flavor structure, that of m ν is given by only three combinations of Next, we impose the unbroken Z 2 symmetry to models and introduce gauge singlet fermions ψ 0 iR as the Z 2 -odd fields. The fermions have Majorana mass terms, We can take the basis where M ψ is diagonalized without loss of generality. For Yukawa interactions of ψ 0 iR with leptons, scalar fields in Table II are also introduced  as Z 2 -odd fields. Scalar fields in Table I and the SM fields are Z 2 -even ones. Then, the lightest Z 2 -odd particle becomes stable. If the lightest Z 2 -odd particle is neutral one, it can be a dark matter candidate. We find that the Majorana neutrino mass matrix can be obtained by the following four kinds of combinations of Yukawa matrices and the weak interaction in addition to the five combinations in eqs. (1)- (5): where Yukawa matrices Y s and Y η are defined in Table II. The Krauss-Nasri-Trodden (KNT) model [12] of m ν at the three-loop level is an example for the combination in eq. (6). The structure in eq. (7) is realized, for example, in the Aoki-Kanemura-Seto (AKS) model [13] at the three-loop level by introducing the Z 2 -odd real singlet scalar boson. Since the three-loop diagram utilizes the scalar interaction with two Higgs doublet fields, the AKS model can explain not only m ν and the dark matter but also the baryon asymmetry of the universe via the electroweak baryogenesis scenario. An 6: The diagram of the fermion line for the combination in eq. (6). Bold red lines are for the  Table II: It is clear that combinations in eqs. (1)-(4) and eqs. (6)-(9) can be classified further to only the following three groups: where symmetric matrices X SR and X SL are given by The matrix X SR is for the effective interactions of right-handed charged leptons while the matrix X SL is for the ones of left-handed leptons. As long as we concentrate on the flavor structure, it seems difficult to discriminate the origin of X SR (X SL ) in eq. (13) (eq. (14)).
We mention here the type-I [15] and the type-III seesaw [16] models, where gauge singlet fermions (for the type-I) or SU(2) L -triplet Majorana fermions (for the type-III) are introduced. The structure of m ν in these models can be included in the Group-III because Yukawa matrices Y A and y ℓ are not used to generate m ν . However, they are exceptions because new scalar fields are not introduced. Discussion in the next section (namely, τ → ℓ 1 ℓ 2 ℓ 3 (ℓ 1 , ℓ 2 , ℓ 3 = e, µ) for the Group-III) is not applicable for these models 5 .

III. DISCUSSION
The neutrino mass matrix m ν is expressed as U * MNS diag(m 1 e iα 12 , m 2 , m 3 e iα 32 )U † MNS , where m i (i = 1-3) are the neutrino mass eigenvalues, α 12 and α 32 are the Majorana phases [61], and U MNS is the Maki-Nakagawa-Sakata (MNS) matrix [62] of the lepton flavor mixing. The 5 There is the box diagram with the W boson and neutral fermions from SU(2) L -singlet or triplet, but the interaction of the neutral fermions with W is suppressed by m ν /M R (the mixing between ν L and the fermions), where M R denotes the fermion mass.
Group-I gives m 1 = 0 or m 3 = 0 because of Det(m ν ) ∝ Det(Y A ) = 0. Although this has been known for the Zee-Babu model [8] (an example of models in the Group-I), our statement is more model-independent. The Group-I is excluded if the absolute neutrino mass is directly measured at the KATRIN experiment [63] whose estimated sensitivity is 0.35 eV at 5 σ confidence level. The indirect bound on the sum of neutrino masses, i m i < 0.23 eV (90% confidence level), was obtained by cosmological observations [64], and sensitivity to i m i = O(0.01) eV is expected in future experiments [65].
The flavor structure of m ν is constrained by the neutrino oscillation data, and the constrained structure can be translated into constraints on the flavor structure (ratios of elements) of X SR of the Group-II and X SL of the Group-III. Hereafter, we denote X SR of the Group-II and X SL of the Group-III as X for simplicity. These interactions can cause the lepton flavor violating (LFV) decays τ → ℓ 1 ℓ 2 ℓ 3 (ℓ 1 , ℓ 2 , ℓ 3 = e, µ). Ratios of the decay branching ratios (BR) of these LFV decays can be determined by the flavor structure of X independently on the overall scale of m ν . In order to evade the strong constraint BR(µ → eee) < 1.0 × 10 −12 [66], LFV decays τ → ℓ 1 ℓ 2 ℓ 3 can be observed at the Belle II experiment [67] only for X ee = 0 or X eµ = 0, which constrains ratios of BR(τ → ℓ 1 ℓ 2 ℓ 3 ) as discussed in the HTM (included in the Group-III) [68]. For X ee = 0 (X eµ = 0), LFV decays τ → ℓee (τ → ℓeµ) do not occur. Since X eℓ elements for the Group-II are enhanced by 1/m e for a given m ν , it is likely that BR(τ → eeµ) for X ee = 0 or BR(τ → eee) for X eµ = 0 is larger than the others. For X ee = X eµ = 0, only τ → eµµ can be observed for the Group-II as shown in the GNR model [10], while τ → µµµ is also possible for the Group-III.
Notice that X ee = 0 for the Group-II and III results in (m ν ) ee = 0, which is excluded if the neutrinoless double beta decay (See e.g. Ref. [69]) is observed or m 3 < m 1 (the inverted mass ordering of neutrinos) is determined by neutrino oscillation experiments (See e.g. Ref. [70]).
Notice also that (X SR ) ee = 0 for the Group-I does not mean (m ν ) ee = 0. Therefore, if (m ν ) ee = 0 is excluded by these neutrino experiments, the observation of τ → ℓee indicates the Group-I because the situation is inconsistent for the Group-II and III.
The discussion above did not require the discovery of new particles. If a charged scalar boson is discovered and dominantly decays into leptons, the branching ratios are expected to be given by Y A (y ℓ ) when the Group-I (II) is assumed. The flavor structure of y ℓ is known, and decays via the y ℓ are dominated by the decay into τ . The flavor structure of Y A is determined by the neutrino oscillation data as For m 1 > m 3 , they are given by Ratios of decay branching ratios BR(s − 1 → eν) : BR(s − 1 → µν) : BR(s − 1 → τ ν) are roughly given by 2 : 5 : 5 for m 1 < m 3 and 2 : 1 : 1 for m 1 > m 3 [71]. Therefore, Group-I and II can be tested by measuring leptonic decays of the charged scalar boson at the collider experiments.
When a group of models is favored by the experiments discussed above, we will try to discriminate models in the group by using details of each model. For example, the doubly-

IV. CONCLUSION
In this letter, we have studied the systematic classification of models for generating Majorana neutrino masses m ν according to combinations of Yukawa interactions. If we use Yukawa interactions for leptons by introducing new scalar fields relevant for these Yukawa interactions, the flavor structure of m ν is given by three combinations: Y s A y ℓ Y s S y ℓ (Y s A ) T , y ℓ (Y s S ) * y ℓ , and Y ∆ S . The Yukawa matrix Y A is antisymmetric while Y s S and Y ∆ S are symmetric. The Yukawa couplings y ℓ are proportional to charged lepton masses. For the case where gauge singlet Z 2 -odd fermions ψ 0 iR and Z 2 -odd scalar fields are additionally introduced, the flavor structure of m ν is determined also by The Yukawa matrices Y s S and Y η S are symmetric, and M ψ is the Majorana mass matrix for ψ 0 iR . Combining these results, we have found that models can be classified into only three groups: m ν ∝ Y s A y ℓ X SR y ℓ (Y s A ) T , y ℓ X * SR y ℓ , and X SL . Here, X SR and X SL are some symmetric matrices. Although the structure of m ν in the type-I seesaw and the type-III seesaw models can be classified in the Group-III, these models are exceptions to the discussion in this letter. Our classification enable us to approach efficiently to the origin of Majorana neutrino masses by testing not each model but each groups of models.
We concentrated on Majorana neutrino masses in this letter. The similar classification of models for Dirac neutrino masses is also desired because the nature may respect the lepton number conservation. This will be presented elsewhere [72].