Texture zeros flavor neutrino mass matrix and triplet Higgs models

One- and two-zero textures for the flavor neutrino mass matrix have been successful in explaining mixing in the neutrino sector. Conservatively, six cases of one-zero textures and seven cases of two-zero textures are compatible with observations. We show that only one case may be most natural in the one- and two-zero textures scheme if the tiny neutrino masses are generated by the type-II seesaw mechanism in the triplet Higgs models.


I. INTRODUCTION
The origin of the tiny masses and flavor mixing of neutrinos is a long-term mystery in particle physics. The seesaw mechanism is the one of the leading theoretical mechanisms to generate tiny neutrino masses. There are three types of seesaw mechanism [1].
3. Type-III: The triplet fermions are introduced in the standard model [13].
To solve the origin of the flavor mixings of neutrinos, there have been various discussions on the texture zeros approach for flavor neutrino masses [14]. In this approach, we assume that the flavor neutrino mass matrix has zero elements.
In the one-zero textures scheme, there are following six cases for the flavor neutrino mass matrix G 1 : All six cases of one-zero textures are consistent with observations [15]. In the two-zero textures scheme, there are fifteen possible combinations of two vanishing independent elements in the 3 × 3 Majorana flavor neutrino mass matrix. The * Electronic address: teruyuki@tokai-u.jp neutrino oscillation data allows only seven out of the fifteen cases [16][17][18][19][20] A 1 : If the neutrino less double beta decay is observed in the future experiments, the A 1 and A 2 cases should be excluded [21,22]. Moreover, Singh shows only B 2 and B 4 are compatible with recent data at 2σ [23]. In this paper, all seven cases of two-zero textures in Eq.(2) are included in our study in a conservative manner.
In this paper, we demonstrate that all six cases of onezero textures (G 1 , G 2 , · · · , C 6 ) and all seven cases of twozero textures (A 1 , A 2 , B 1 , · · · , B 4 , C) are excluded if the following two conditions are satisfied: C1: Neutrino masses are generated by the type-II seesaw mechanism in the triplet Higgs models.
Moreover, we show that only G 6 case is viable if the condition C1 as well as the following conditions are satisfied: C3: All of three lepton flavor violating processes µ → eee, τ →μµµ and τ →ēee are observed experimentally or are undoubtedly predicted theoretically.
Even if the part of these three lepton flavor violating processes is allowed such as BR(µ →ēee) = 0, BR(τ → µµµ) = 0 and BR(τ →ēee) = 0, other cases of one-and two-zero textures may be allowed; however, we show that only G 6 case may be most natural in one-and two-zero textures scheme. The paper is organized as follows. In Sec.II, we present a brief review of the triplet Higgs model. In Sec.III, we show only G 6 case may be most natural in the one-and two-zero textures scheme if the neutrino masses are generated by the type-II seesaw mechanism in the triplet Higgs models. Section IV is devoted to a summary.

II. TRIPLET HIGGS MODEL
We assume that neutrino masses are generated by the type-II seesaw mechanism in the triplet Higgs models. In the triplet Higgs models [7][8][9][10][11][12], an SU (2) triplet scalar field is introduced into the particle contents of the standard model. This triplet of scalar fields yield a Majorana mass of the neutrinos via the following Yukawa interaction: where y ij (i, j = e, µ, τ ) is the (i, j) element of the complex and symmetric Yukawa coupling matrix, C is the charge conjugation, τ 2 is a Pauli matrix, and ψ iL = (ν i , ℓ i ) T L is a standard model left-handed lepton doublet. After ξ 0 develops a nonzero vacuum expectation value v ∆ = ξ 0 , Majorana neutrino masses M ij are generated.
One of the most important relations in the triplet Higgs models is the one-to-one correspondence between the flavor neutrino masses M ij and Yukawa couplings y ij [48][49][50]: These Yukawa matrix elements y ij are also related with lepton flavor violating processes [48][49][50][51]. For example, the virtual exchange of doubly charged Higgs bosons induces an effective interaction of four charged leptons for ℓ m →l i ℓ j ℓ k decay at tree level. The branching ratio for the lepton flavor violating decays µ →ēee and τ →l i ℓ j ℓ k are given by and where S = 1(2) for j = k (j = k), G F is the Fermi coupling constant and M ±± denotes the mass of the doubly charged Higgs bosons [50].
Thanks to the one-to-one correspondence between the flavor neutrino masses and Yukawa couplings, the branching ratios of the lepton flavor violating decay ℓ m →l i ℓ j ℓ k directly connect with the neutrino flavor masses: In the next section, we use these lepton flavor violating processes µ →ēee, τ →μµµ, τ →ēee to test availability of the one-and two-zeros textures.

III. TEXTURE ZEROS
In this section, we assume that all of three lepton flavor violating processes µ →ēee, τ →μµµ and τ →ēee are explicitly forbidden experimentally or theoretically.
In this case, at least the branching ratio BR(µ →ēee) as well as M ee and/or M eµ should vanish. If we require the conditions of M ee = 0 and/or M eµ = 0 to the G 3 case in the one-zero textures scheme: the following three flavor neutrino mass matrix are ob- however, one-zero textures assumption is violated in these matrices by an additional vanishing entry. Therefore, the G 3 case in the one-zero textures scheme should be excluded if the lepton flavor violating process µ →ēee is explicitly forbidden. In the same manner, we can exclude the following G 4 ,G 5 and G 6 cases if the lepton flavor violating process µ →ēee is explicitly forbidden. Moreover, the following B 1 , B 4 and C cases in the two-zero textures scheme: are also excluded if we require the conditions of M ee = 0 and/or M eµ = 0 (two-zero textures assumption should be violated with this requirement) Consequently, the G 3 ,G 4 ,G 5 and G 6 cases of one-zero textures and B 1 , B 4 and C cases of two-zero textures should be excluded if the lepton flavor violating process µ →ēee is explicitly forbidden.
Addition to the lepton flavor violating process µ → eee, we can use other two lepton flavor violating processes τ →μµµ and τ →ēee to test compatibility of the oneand two-zero textures. Table I shows the compatibility of the cases in the one-and two-zero textures scheme with the vanishing branching ratios BR(µ →ēee) = 0, BR(τ →μµµ) = 0 and BR(τ →ēee) = 0. The symbol × means the corresponding case should be excluded.
We conclude that all six cases of one-zero textures (G 1 , G 2 , · · · , C 6 ) and all seven cases of two-zero textures (A 1 , A 2 , B 1 , · · · , B 4 , C) should be excluded if the neutrino masses are generated by the type-II seesaw mechanism in the triplet Higgs models and all of three lepton flavor violating processes µ →ēee, τ →μµµ and τ →ēee are explicitly forbidden.
In this section, we assume that all of three lepton flavor violating processes µ →ēee, τ →μµµ and τ →ēee are observed experimentally or are undoubtedly predicted theoretically.
In this case, at least the branching ratio BR(µ →ēee) as well as M ee and M eµ cannot vanish. The nonvanishing elements M ee and M eµ (M ee = 0 and M eµ = 0) are inconsistent with the G 1 ,G 2 ,A 1 ,A 2 ,B 2 and B 3 cases in II: Compatibility of the cases in the one-and twozero textures scheme with the nonvanishing branching ratios BR(µ →ēee) = 0, BR(τ →μµµ) = 0 and BR(τ →ēee) = 0. The symbol × means the corresponding case should be excluded.
the one-and two-zero textures scheme: Therefore, the G 1 ,G 2 ,A 1 ,A 2 ,B 2 and B 3 cases in the oneand two-zero textures scheme should be excluded if the lepton flavor violating processes µ →ēee are observed experimentally or are undoubtedly predicted theoretically.
Addition to the lepton flavor violating process µ → eee, other two lepton flavor violating processes τ →μµµ and τ →ēee are available for evaluation of the viability of the one-and two-zero textures. Table II shows the compatibility of the cases in the one-and two-zero textures scheme with the nonvanishing branching ratios BR(µ → eee) = 0, BR(τ →μµµ) = 0 and BR(τ →ēee) = 0. The symbol × means the corresponding case should be excluded.
We conclude that only G 6 case is viable in one-and two-zero textures of the flavor neutrino mass matrix if the neutrino masses are generated by the type-II seesaw mechanism in the triplet Higgs models and all of three lepton flavor violating processes µ →ēee, τ →μµµ and τ →ēee are observed experimentally or are undoubtedly predicted theoretically. III: Allowed cases in the one-and two-zero textures scheme. The symbol "NZ" means some nonzero values of the branching ratios.

C. Hybrid cases
Based on the above discussion, it turned out that if the neutrino masses are generated by the type-II seesaw mechanism in the triplet Higgs models and all of three lepton flavor violating processes µ →ēee, τ →μµµ and τ →ēee are forbidden, there is no room for one-and two-zero textures. On the other hands, if all of these three processes exist, only G 6 case is viable in one-and two-zero textures.
If the part of these three lepton flavor violating processes is allowed such as other cases of one-and two-zero textures may be allowed. For example, in the case shown in Eq.(14), the G 6 case is ruled out and only B 1 case is allowed. Table III shows the allowed cases in the one-and two-zero textures scheme. The symbol "NZ" means some nonzero values of the branching ratios. It is remarkable that the each of G 1 , G 2 , · · · , C cases appears only once in the Table III. Therefore, we can predict the allowed combination of nonvanishing branching ratios by the one-and two-zero flavor neutrino mass matrix textures. Although the problem of whether three lepton flavor violating processes µ →ēee, τ →μµµ and τ →ēee are forbidden or not is unsolved yet, we can suggest that either BR(µ →ēee) = BR(τ →μµµ) = BR(τ →ēee) = 0, (15) or BR(µ →ēee) = BR(τ →μµµ) = BR(τ →ēee) = 0, may be most natural case. Otherwise, the appropriate selection mechanisms for ℓ m →l i ℓ j ℓ k decay at tree level are required in the models. We can conclude that if the tiny neutrino masses are generated by type-II seesaw mechanism, only G 6 case may be most natural in one-and two-zero textures scheme. This is the main result in this paper.

D. Numerical calculations
Although the main result in this paper has already obtained in subsection III C, an additional numerical study may be required to improve our discussions. According to the conclusion in subsection III C, only G 6 case may be most natural in one-and two-zero textures scheme if the tiny neutrino masses are generated by type-II seesaw mechanism. In this subsection, we present a phenomenology for the G 6 case.
First we show brief reviews of the neutrino mixings, useful relations for the one-zero textures and observed data from neutrino experiments as a preparation of our numerical calculations. Then, we show a prediction for the G 6 case.
Neutrino mixings: The flavor neutrino mass matrix M is related with the diagonal neutrino mass matrix where m i (i = 1, 2, 3) is a neutrino mass eigenstate and denotes the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix [52][53][54][55]. We used the abbreviations c ij = cos θ ij and s ij = sin θ ij (i, j=1,2,3) where θ ij is a neutrino mixing angle. The Dirac CP phase is denoted by δ and the Majorana CP phases are denoted by α 1 and α 2 .
Useful relations for one-zero textures: The requirement of M ij = 0 in the one-zero textures yields where This condition reads to (for examples, see Refs. [34,56]) and The ratio of two squared mass differences is given by where the squared mass difference is defined by ∆m 2 ij = m 2 i − m 2 j . Eqs. (22), (23) and (24) are useful when we search the allowed parameter sets under the requirement of M ij = 0.
Observed data: Although the neutrino mass ordering (either so-called normal mass ordering m 1 m 2 < m 3 or inverted mass ordering m 3 < m 1 m 2 ) is not determined, a global analysis shows that the preference for the normal mass ordering is mostly due to neutrino oscillation measurements [21,57]. Upcoming experiments for neutrinos will be solve this problem [58]. In this paper, we assume the normal mass hierarchical spectrum for the neutrinos.
A global analysis of current data shows the following the best-fit values of the squared mass differences and the mixing angles for the normal mass ordering [59]: where the ± denote the 1σ region and the parentheses denote the 3σ region. Moreover, the following constraints from the cosmological observation of the cosmic microwave background radiation [21,60] as well as from the neutrino less double beta decay experiments [21,22] are obtained.
A phenomenology for G 6 case: Now, we show a prediction for the G 6 case by numerical calculations.
In our numerical calculation, we require that the square mass differences ∆m 2 ij , mixing angles θ ij and the Dirac CP violating phase δ are varied within the 3σ experimental ranges, the Majorana CP violating phases α 1 and α 2 are varied within their full possible ranges and the lightest neutrino mass is varied within 0.01 − 0.1 eV. We also require that the constraints |M ee | < 0.155 eV and m i < 0.241 eV (TT, TE, EE+LowE+lensing [23,60]) are satisfied. We estimate the ratio as one of the predictions of the one-zero textures.
We show an example of the results of our numerical calculations for the G 6 case. A point set  Figure 1 shows that the prediction of R = BR(τ → µµµ)/BR(τ →ēee) for the lightest neutrino mass m 1 for the G 6 case. Currently, we have only upper limit of BR(τ →μµµ) < 2.1 × 10 −8 and BR(τ →ēee) < 2.7 × 10 −8 from observations [55]. If these branching ratios are decided in the future experiments, R ≃ 0.6−812 supports to the G 6 case within the type-II seesaw generation of the neutrino masses in the triplet Higgs models.

IV. SUMMARY
One-and two-zero textures for the flavor neutrino mass matrix have been successful in explaining mixing in neutrino sector. In this paper, we have shown that all cases of one-and two-zero textures are excluded if the tiny neutrino masses are generated by type-II seesaw mechanism in the triplet Higgs models and all of three lepton flavor violating processes µ →ēee, τ →μµµ and τ →ēee are explicitly forbidden experimentally or theoretically. We have also shown that if all of these three lepton flavor violating processes exist, only G 6 case is viable within the one-and two-zero textures.
Even if the part of these three lepton flavor violating processes is allowed such as BR(µ → 3e) = 0, BR(τ → 3µ) = 0 and BR(τ → 3e) = 0, we can suggest that the most natural case is either BR(µ → 3e) = BR(τ → 3µ) = BR(τ → 3e) = 0 or BR(µ → 3e) = BR(τ → 3µ) = BR(τ → 3e) = 0. Otherwise, the appropriate selection mechanisms for ℓ m →l i ℓ j ℓ k decay at tree level are required in the models. Therefore we have concluded that if the tiny neutrino masses are generated by the type-II seesaw mechanism in the triplet Higgs models, only G 6 case may be most natural in one-and two-zero textures scheme.
Finally, a prediction for G 6 case has been shown. The ratio R = BR(τ →μµµ)/BR(τ →ēee) should be R ≃ 0.6 − 812 for the G 6 case within the type-II seesaw generation of the neutrino masses in the triplet Higgs models.