More five-parameter models for lepton mixing

We propose four new lepton-mixing textures that may be enforced through well-defined symmetries in renormalizable models. Each texture has only five parameters to predict nine observables; the textures are therefore able to produce testable predictions for various neutrino mass observables and for the CP-violating phase delta. The models are based on the type-I seesaw mechanism; their charged-lepton mass matrices are diagonal because of the symmetries imposed. Each model has three versions, depending on the identification of the charged leptons. Testing all the models, we have found that five of them agree with the data at the 1 sigma level when the neutrino-mass ordering is normal, and two models agree with the data for an inverted ordering. We detail the predictions of each of those seven models.


Introduction and notation
In this paper we use the type-I seesaw mechanism [1] for suppressing the light-neutrino masses. Let L and R be 3 × 1 column matrices that subsume the three left-handed and the three right-handed, respectively, charged-lepton fields; let ν L and ν R analogously subsume the three left-handed and the three right-handed neutrino fields. The lepton mass terms are given by where C is the charge-conjugation matrix in Dirac space. We have added to the Standard Model three right-handed neutrinos with Majorana mass terms subsumed by the 3 × 3 symmetric matrix (in flavour space) M R . In all the models in this paper the charged-lepton mass matrix M is diagonal: where |a α | = m α for α = e, µ, τ . The neutrino Dirac mass matrix M D is also diagonal in all our models: The seesaw mechanism takes place when the matrix M R is invertible and its eigenvalues are much larger than the |b α |. One then obtains an effective light-neutrino Majorana mass matrix where [2] M (1) We shall use the approximation M ν = M (1) ν . The diagonalization of M ν proceeds as where m 1,2,3 are the light-neutrino masses; they are non-negative real. Since the chargedlepton mass matrix is diagonal from the start, U in equation (6) is the lepton mixing matrix. We use the parameterization in ref. [3]: where ≡ s 13 exp (iδ), c ij = cos θ ij , and s ij = sin θ ij for ij = 12, 23, 13. Three different groups of phenomenologists [4,5,6] have derived, from the data provided by various neutrino-oscillation experiments, values for the mixing angles θ 12,23,13 , for the phase δ, and for the neutrino squared-mass differences.
In general the matrix M ν determines nine observables: the three neutrino masses, the three mixing angles, the Dirac phase δ, and the Majorana phases α 21 and α 31 . If M ν contains less than nine independent rephasing-invariant parameters (IRIP)-i.e., quantities that are invariant under (M ν ) αβ → (M ν ) αβ exp [i (ξ α + ξ β )], where the three phases ξ e,µ,τ are arbitrary-then there will be some relations (sometimes called 'sum rules') among the nine observables. This happens in particular when M ν has two 'texture zeroes': if two out of the six independent matrix elements of M ν vanish, then there are only five IRIP in M ν , 1 hence four sum rules among the nine observables. Seven viable two-texture-zero cases have been identified in ref. [7]. 2 Other viable cases-or sometimes full models-in which M ν contains only five IRIP have been discovered, for instance, in refs. [8] and [9].
Therefore, it may in general be parameterized by three complex parameters r, s, and t: The matrix A is useful for us because in the models of this paper M D is diagonal, cf. equation (3).
Thus, if M R possesses some symmetry, then that symmetry gets transferred to M ν via A.
In section 2 we shall present models 1 and 3. In section 3 we shall present models 4 and 5. Since equations (12b) are the same as equations (12a) after a µ-τ interchange, and since equations (12f) and (12g) are the same as equations (12d) and (12e), respectively, after an e-µ interchange, our models 1, 4, and 5 can also be identified as models 2, 6, and 7, respectively, if one labels the charged leptons in a different manner. An analysis of the practical consequences of our sum rules is deferred to section 4; it turns out that models 1-5 agree with the data at the 1σ level when the neutrino mass ordering is normal ('NO'), viz. m 1 < m 2 < m 3 , while models 6 and 7 agree with the data at the 1σ level when the neutrino mass ordering is inverted ('IO'), viz. m 3 < m 1 < m 2 . A short summary of our findings is attempted in section 5.

Models 1 and 3
The models in this section are inspired by those in ref. [11], viz. they are based on the idea of a (leading-order) antisymmetry of M R under an e-µ interchange. The models in ref. [11] either had four, five, or six IRIP in M ν ; the models with either four or five IRIP were unable to fit the data, only the models with six IRIP agreed with them-but those models had many IRIP, hence little predictive power. The models that we construct here only have five IRIP and are nonetheless successful.
All the models in this paper have gauge group SU (2) × U (1). There are three left-handed-lepton gauge-SU (2) doublets D α = (ν αL , α L ) T , three right-handed chargedlepton SU (2) singlets α R , and three right-handed-neutrino gauge singlets ν αR . In all the models in this paper we use two scalar gauge-SU (2) doublets φ 1 and φ 2 . 3 Let v a (a = 1, 2) denote the vacuum expectation values (VEVs) of the neutral components φ 0 a of In the models in this section there is one complex scalar gauge singlet S. We introduce the flavour-lepton-number symmetries L α ; the dimension-four terms in the Lagrangian respect those symmetries but lower-dimension terms are allowed to break them softly. The multiplets D α , α R , and ν αR have U (1) charge +1 under L α and U (1) charges 0 under the L β with β = α. We also enforce a Z 4 symmetry that interchanges e and µ: The Yukawa Lagrangian coupling the leptons to the scalar doublets is therefore Therefore, the charged-lepton mass matrix and the neutrino Dirac mass matrix are diagonal as anticipated in equations (2) and (3), respectively, with 4 The doublet φ 2 and its Yukawa couplings in lines (14b) and (14d) are needed so that m e = m µ and b e = b µ . There are right-handed-neutrino Majorana mass terms The terms in L Mν violate the family-lepton-number symmetries L α ; this is allowed because those terms have mass dimension three.
Model 1: In this model the singlet S has L e = L µ = +1 and L τ = 0. There is then a coupling where y s is a Yukawa coupling constant. The Majorana mass matrix of the right-handed neutrinos is where w is the VEV of S. 5 Therefore, the matrix A is given by equation (12a) with r = m /(2y s w). 4 Since |a e | = m e |a µ | = m µ , there is a finetuning making y 3 v 2 ≈ −y 2 v 1 . This finetuning may be justified through the introduction of an additional symmetry in the model [12]. We shall not pursue that idea here. 5 We assume that y s w is of the same order of magnitude as m and m .

Model 3:
In this model the singlet S has L e = L µ = 0 and L τ = +2. There is a coupling Then, The matrix A is as in equation (12c) with r = m 2 (my s w).

Models 4 and 5
The models in this section are inspired by the ones in ref. [9]. They use two complex scalar gauge singlets S 1 and S 2 and one real singlet S 3 ; thus, their scalar sector is larger than the one of the models of the previous section. In models 4 and 5 we do not use flavour-lepton-number symmetries, rather we employ a Z 8 symmetry to the same effect. Let σ = exp (iπ/4), then the Z 8 symmetry is given by This symmetry Z 8 allows for the Yukawa Lagrangian The charged-lepton mass matrix and the neutrino Dirac mass matrix are given by equations (2) and (3), respectively, with The symmetry Z 8 also allows a bare Majorana mass term The Majorana mass matrix of the right-handed neutrinos is then where w k = 0 |S k | 0 for k = 1, 2, 3. Note that w 3 is real because S 3 is a real scalar field. The potential of the scalar singlets is , (26b) but in both model 4 and model 5 there are additional symmetries (see below) that enforce We parameterize where w is non-negative and θ ∈ [0, π]. Then, where ψ = ψ 2 − ψ 1 .

Model 4
In model 4 there is additionally the Z 2 symmetry The symmetry (30) does not eliminate the bare mass term (24); in the Yukawa Lagrangian (22) it makes y 3 = y 2 , y 4 = 0, y 6 = −y 5 , (31a) y 9 = y 8 , y 10 = 0, y 12 = −y 11 , (31b) y 14 = y 13 , y 16 = y 15 . (31c) Because of equations (31c), we now have instead of equation (25). If one assumes w 1 = w 2 , then one obtains equations (12d) with 2r = −y 15 w 3 /(m − y 15 w 3 ). In order to justify the assumption w 1 = w 2 , one must look at the potential of the scalar singlets. Because of the symmetry (30), equations (27) hold and moreover bothm and λ 7 are real. In equation (29) this leads to where It can readily be checked that the minimum of V 0 in equation (33) is obtained for θ = π/2 and ψ = 0, i.e. for w 1 = w 2 , provided B < 0, C < 0, and A < −C. Thus, there is a range of the parameters of the potential for which the desired minimum is obtained. At low energy, this minimum will be disturbed by terms in the potential that involve the doublets φ a , in particular the term φ † However, those terms are of order v 2 /w 2 1 relative to the potential in equation (26), where v is the electroweak scale and w is assumed to be at the seesaw scale. One may neglect those terms in just the same way as one neglects M (2) ν when compared to M

Model 5
Instead of the Z 2 symmetry of (30), in model 5 we employ the CP symmetry 6 where x = (t, r) andx = (t, − r). In the Lagrangian (22), the CP symmetry (35) enforces Moreover, in equation (24) m becomes real. One then has If one now assumes |w 1 | = |w 2 |, then one obtains equation (12e) with In the scalar potential (26), equations (27) hold because of the CP symmetry (35). Thus, V 0 is as in equation (29). There is a stability point of V 0 for θ = π/2, i.e. |w 1 | = |w 2 |; we assume that that stability point is the absolute minimum of V 0 .
4 Confrontation with the phenomenological data 4.

Introduction
We have tested the four sets of conditions (α = β = γ = α) against the phenomenological data [4,5,6], both for the three choices of α (e, µ, or τ ) and for the two choices of neutrino mass ordering (NO or IO). Thus, we have tested 12 different models and, for each of them, two mass orderings. We have found that, out of the 24 possibilities, seven models and mass orderings are viable-in the sense that we shall explain below-viz. models 1-5 for NO and models 6 and 7 for IO, cf. the listing (12).
In this section we study in some detail the predictions of each of those models for the Dirac phase δ and for the neutrino mass observables, viz. the mass of the lightest neutrino m minimum (m minimum = m 1 for NO and m minimum = m 3 for IO), the total mass of the light neutrinos the mass relevant for neutrinoless double-beta decay and the mass relevant for standard β decay We recall the cosmological bound [13] m ν < 0.12 eV, which turns out to be relevant in constraining models 4 and 5, but not the other five models.
We have used as input the nine observables δ, α 21 , α 31 , s 2 12 , s 2 13 , s 2 23 , m minimum , ∆m 2 solar ≡ m 2 2 − m 2 1 , and ∆m 2 atmospheric . (Following ref. [6], we define ∆m 2 atmospheric = m 2 3 − m 2 1 > 0 for NO and ∆m 2 atmospheric = m 2 3 − m 2 2 < 0 for IO.) For each set of input observables, we have computed firstly M ν = U * × diag (m 1 , m 2 , m 3 ) × U † and secondly the A-matrix elements A αβ = (M ν ) αβ (M −1 ν ) βα . We have numerically generated thousands of sets of input observables that reproduce each of our constraint equations (39) with extremely great accuracy. 7 We have firstly tested our models in the following way. We have searched for sets of input observables such that all six observables s 2 12 , s 2 13 , s 2 23 , δ, ∆m 2 solar , and ∆m 2 atmospheric are inside their respective 1σ Confidence Level (CL) intervals for any one of the three phenomenological fits [4,5,6]. If we were able to satisfy the constraints of one of our models and mass orderings through observables fully inside the 1σ ranges of one of the phenomenological fits, then we have classified that model and mass ordering as viable.
To be explicit, we have found that both models 1 and 3 for NO and models 6 and 7 for IO can be met through input observables inside the 1σ intervals of either ref. [4], ref. [5], or ref. [6]; while models 4 and 5 with NO can be satisfied within the 1σ domains of both ref. [4] and ref. [5]; finally, model 2 with NO can be realized at 1σ CL through ref. [5]. All other models and mass orderings cannot be reproduced with 1σ CL input through any of the three phenomenological fits; therefore we have discarded them. After this choice of viable models, we have proceeded to analyze each model in more detail. We have followed in this endeavour ref. [14] and we have used exclusively the phenomenological fit of ref. [6]. 8 In ref. [6], the χ 2 profiles of s 2 23 and δ are not symmetrical relative to the best-fit values; moreover, those two observables are correlated with each other much more strongly than (with) the other four oscillation observables. It makes therefore sense to treat s 2 23 and δ differently from the remaining input. The input values of the observables never coincide exactly with the best-fit values; in order to measure the agreement with phenomenology of each of our 'points', i.e. sets of input observables, we have used a function χ 2 = χ 2 (1) + χ 2 (2) + χ 2 (3) . Here, accounts for the fact that the overall quality of the phenomenological fit is poorer for IO than for NO. The number 4.71254 is the minimum value of the quantities ∆χ 2 (X) (where X is successively s 2 12 , s 2 13 , ∆m 2 solar , and ∆m 2 atmospheric ) depicted in the blue curves of figure 1 of ref. [6]. Because of χ 2 (3) , most fits with IO are of much worse absolute quality than fits with NO, in particular our models 6 and 7 fit the data much worse than models 1-5.
For a more efficient sampling of the space of input parameters, we have used global minimization algorithms and we have performed the minimization of χ 2 for each input point. Specifically, we have minimized χ 2 = χ 2 model + χ 2 (1) for various fixed s 2 23 and δ. In this way, at each point in the s 2 23 -δ plane we have the minimum relative to all the other oscillation parameters. In order to find the exact value of χ 2 minimum , we have added ∆χ 2 (s 2 23 ) and ∆χ 2 (δ) to χ 2 (1) , because it is much easier to include one-dimensional interpolations of ∆χ 2 (s 2 23 ) and of ∆χ 2 (δ) in a FORTRAN code than to include a two-dimensional interpolation of ∆χ 2 (s 2 23 , δ). Later, we have recalculated all the discovered input parameters by using MATHEMATICA with a two-dimensional interpolation of ∆χ 2 (s 2 23 , δ).

Models 1-5
Now look at figure 1. Each row of that figure corresponds to the model that is defined by the conditions that are written in the top-left corner of the left panel of the row, viz. to models 1, 2, 3, and 4, respectively. All these models are for a normal ordering of the neutrino masses, thus m minimum = m 1 . In figure 1, just as in figures 3 and 4, we do not display any panels corresponding to model 5, because the predictions of models 4 and 5 are almost identical to each other.
In the left panels of figure 1 one sees, in different shades of blue, the 1σ, 2σ, and 3σ CL regions in the s 2 23 -δ plane that are allowed by the phenomenological data of ref. [6]. The stars mark the best-fit value of (s 2 23 , δ). The blue regions in the left panels are identical in all four rows of figure 1. The red regions in those panels are specific to each model; they consist of points that (1) − χ 2 (1),minimum ≤ 11.83, where χ 2 (1),minimum is the smallest value of χ 2 (1) in each region of red points; χ 2 (1) − χ 2 (1),minimum ≤ 11.83 corresponds to the 3σ CL for a Gaussian distribution with two degrees of freedom (in this case, s 2 23 and δ). would be able to be much lower, as shown by the dashed red lines in that panel. Another interesting feature of model 4 is a large forbidden zone in the s 2 12 -s 2 23 plane; that zone, with low s 2 12 and high s 2 23 , can be observed in the right panel of figure 2. In the right panels of figure 1 one sees, for each model 1-4, the points that have χ 2 − χ 2 minimum smaller than 2.3 (1σ or 68.27% CL), 6.18 (2σ or 95.45% CL), and 11.83 (3σ or 99.73% CL). In drawing the right panels we have used the full function χ 2 = χ 2 (1) + χ 2 (2) + χ 2 (3) instead of just χ 2 (1) like in the left panels. It should be stressed that, even though all four right panels of figure 1 have a light-blue-coloured zone corresponding to χ 2 − χ 2 minimum < 2.3, that does not mean that all four models 1-4 fit the data equally well, because χ 2 minimum is different for the four models. The values of χ 2 minimum are given in the last row of table 1; they make clear that models 1 and 3 agree with the data almost perfectly, while model 2 is not quite as good and model 4 (and also model 5) is even worse. For instance, all the points with χ 2 − χ 2 minimum < 2.3 for model 1 have χ 2 < 2.7 and are therefore better than even the best point of model 4.
One sees in figure 1 that both models 1 and 3 display two different 'solutions', one of them with δ ∼ −120 • and the other one with δ ≈ 120 • . Under complex conjugation of the lepton mixing matrix, i.e. under δ → −δ, α 21 → −α 21 , and α 31 → −α 31 the conditions defining each model remain invariant, but the phenomenological bounds on δ do not; this is the reason why, for instance for model 1, there are two solutions with symmetric values of the phases-but one of those solutions has much higher values of χ 2 − χ 2 minimum . For model 3 all the points of the second solution have χ 2 − χ 2 minimum > 9 and therefore that solution does not appear in table 1.
Comparing the left and right panels of figure 1, one sees that all models 1-3 severely constrain the phase δ, but they do not constrain s 2 23 by themselves alone. Model 4 has s 2 23 correlated with m ν and, even when m ν becomes very large (i.e. when the light neutrinos are almost degenerate), s 2 23 0.418 is constrained; after the addition of the cosmological bound (43) the constraint becomes much stronger. Model 4 also restricts s 2 12 , see the right panel of figure 2. Next look at figure 3. There, one sees the same regions as in the right panels of figure 1, but now displaying the Majorana phases α 21 and α 31 , and also the smallest neutrino mass m 1 , against δ. Only points that comply with the cosmological bound on m ν are displayed; this is not an effective constraint for models 1-3, but it severely constrains model 4 (and model 5).
In figure 4 one observes the predictions of each model 1-4 for the mass parameters m ν and m ββ . The pink areas in figure 4 are the same for all models and they are allowed by the phenomenological constraints only; the areas in various shades of blue are allowed at the 1σ, 2σ, and 3σ CL by the phenomenological constraints together with each model's conditions. One sees that each model strongly constrains the mass parameters,  Table 1: The 3σ bounds for various observables in the models with normal neutrino mass ordering. These bounds correspond to χ 2 − χ 2 minimum ≤ 9, which is equivalent to 3σ CL for one degree of freedom; they take into account the cosmological bound m ν < 120.0 meV [13]. For model 5 the values are the same as for model 4, with the exceptions 10 × s 2 23 (5.48 to 6.14), δ (154 • to 213 • ), α 31 (−36 • to 46 • ), and χ 2 minimum (3.82).
restricting them to a much smaller range than the one allowed by phenomenology only. It was already clear from the right panels of figure 3 that models 1 and 2 work for much lower values of the neutrino masses than models 3 and 4; m 1 ∼ 10 meV for models 1 and 2 while m 1 ∼ 25 meV for models 3 and 4. This same fact is observed in figure 4, where m ν -but not m ββ , which includes some interference effects-is much higher in models 3 and 4 than in models 1 and 2. Notice that a large otherwise-allowed range of model 4 has been eliminated by the cosmological bound on m ν ; the same may soon happen to model 3, which predicts m ν 105 meV. We have summarized in table 1 the predictions of each of the models with NO. In that table we only display points with χ 2 − χ 2 minimum ≤ 9, therefore the ranges are somewhat narrower than the ones observed in the figures, where the 3σ regions have χ 2 − χ 2 minimum ≤ 11.83. For the same reason, the second solution for model 3 does not appear in table 1.
The observables s 2 12 , s 2 13 , ∆m 2 solar , and ∆m 2 atmospheric are not constrained by models 1-5, with the exception s 2 12 ∈ [0.320, 0.350] in models 4 and 5; this is not, however, because of the models themselves, but rather because of the cosmological bound, that leads those models to necessitate both a rather high s 2 23 and a rather high s 2 12 .

Junction of models 4 and 5
Models 4 and 5 have almost the same predictions and, as a matter of fact, we may join them in only one model, defined by This model agrees with experiment and has χ 2 minimum = 4.22, which is not much worse than either model 4 or model 5 separately. The CP-violating phases are eliminated: δ = α 21 = π and α 31 = 0, rendering this model CP-conserving in the leptonic sector. The predictions for the mass observables and for s 2 23 are exactly the same as the ones displayed in table 1 for model 4.
The plus of this model is that it provides a clear-cut correlation among s 2 12 , s 2 23 , and m ν . That correlation is displayed in figure 5. On the other hand, this model requires both s 2 12 and s 2 23 to be quite above their best-fit values; that is the reason why χ 2 minimum is rather high for this model. In figure 8 one sees the predictions of the two IO models for the mass parameters. One observes once again the great difference between the two solutions of model 7, with one of them producing a much lower m ββ than the other one. It is interesting to observe that both models admit m ββ ∼ 49 meV, which is much higher than in the models with NO.

Summary and conclusions
In this paper we have shown that four new types of constraints on the lepton mass matrices, given in equations (39), can be derived through adequate symmetries imposed on renormalizable models furnished with three right-handed neutrinos and a type-I seesaw mechanism. Each of those constraints leads to a Majorana neutrino mass matrix with only five independent rephasing-invariant parameters and, therefore, predictive power for the CP-violating phase δ and for various neutrino-mass quantities. That predictive power has been studied in some detail in section 4 of the paper, especially taking into account the correlations between δ and the mixing anle θ 23 displayed by the phenomenological data of ref. [6]. We have found that a total of seven models are able to fit the data at the 1σ level for at least one of the three phenomenological papers [4,5,6]. The predictions of each of our models have been given in tables 1 and 2.               Figure 8: The predictions of models 6 (top row) and 7 (the other two rows) for the sum of the light-neutrino masses and for the mass parameter responsible for neutrinoless 2β decay. The pink areas are the ones allowed by phenomenology alone, for an inverted ordering of the neutrino masses; the blue areas include the constraints of each model. The right panels are zooms of the marked areas in the left panels.