CP violation in neutrino mixing with $\delta = -\pi/2$ in $A_4$ Type-II seesaw model

We study a class of models for neutrino mass matrix in Type-II seesaw with $A_4$ family symmetry. The resulting neutrino mass matrix can be naturally made to respect a $\mu-\tau$ exchange plus CP conjugate symmetry (GLS) with the CP violating phase $\delta$ and the mixing angle $\theta_{23}$ predicted to be $\pm\pi/2$ and $\pi/4$, respectively. When GLS is explicitly broken by complex Yukawa couplings, the model predictions for $\delta$ and $\theta_{23}$ can be significantly modified. Should future experiments will indeed determine $\theta_{23}$ and $\delta_{CP}$ away from the GLS limit values, one then had to consider models with broken GLS. We study several simple scenarios to show how the modifications arise when GLS is broken and how future experiments can test this class of models.


I. INTRODUCTION
Experiments have collected a lot of precious information about the neutrino mixing parameters. The mixing angles in the Pontecorvo-Maki-Nakagawa-Sakata matrix [1] V P M N S are not always small [2][3][4] as their quark mixing counter part [2,5]. In the standard parametrization [2,6] for three neutrino mixing commonly used [3,4], the mixing angle θ 23 is close to π/4, θ 12 is also large, θ 13 is relatively small but away from zero. There are also evidences showing that the CP violating Dirac phase δ is close to −π/2 (or 3π/2). If these data are further confirmed, the neutrino mass matrix will have a simple form. Assuming that neutrinos are Majorana particles, the neutrino mass matrix defined by the term giving neutrino masses in the Lagrangian (1/2)ν L m ν ν c L has the following form, where c ij and s ij are cos θ ij and sin θ ij , respectively. They are all normalized to be positive.
With δ = −π/2 and θ 23 = π/4, m ν has the following form [7,9,10] Note that in the most general case, because non-zero Majorana phases, the parameters a,b, c, d, β and γ are all complex. One has the degrees of freedom to redefine the neutrino fields phases and the most general form of the above mass matrix can be rewritten as where the phases p i are arbitrary. One can choose some particular values for p i to obtain forms of m ν for convenience of analysis. For example the "-" sign for the "13" and "31" entries can be removed by choosing p 1 = p 2 = 0 and p 3 = π, the resultant matrix can be written in a more familiar forms If neutrinos do not have non-trivial Majorana phases (but mass can be positive or negative), the mass matrix has the following form Replacing δ = −π/2 by δ = π/2, the neutrino mass matrix is given in a similar form as that in eq.(6), but β and γ need to be multiplied by a "-" sign. This implies that without further information given, a general mass matrix in the form given by eq.(6) can give δ = ±π/2 and θ 23 = π/4. Whether they predict +π/2 or −π/2, additional information need to be provided [11]. It has been pointed out that the general form in eq.(6) is a necessary condition for δ = ±π/2 and θ 23 = π/4, but not sufficient condition. In our later discussions, unless specified, the mass matrix of the form in eq.(6) is always referred to a general form whose elements are not necessarily given by those in eq.(4). While the mass matrix in the form of eq.(7) provide sufficient condition for δ = ±π/2 and θ 23 = π/4 when s 13 and sin δ are not zero. The simplicity of the above mass matrix may serve as a good starting point to understand possible underlying theory. In fact it has been shown that the above neutrino mass matrix is a consequence of imposing a symmetry of the form e → e, µ and τ exchange with CP conjugation discussed by Grimus and Lavoura in Ref. [8], which we will refer to as the Grimus-Lavoura symmetry (GLS).
In this work, we study realizations of δ = ±π/2 and θ 23 = π/4 in type-II seesaw model with A 4 flavor symmetry. Models based on A 4 symmetry has been shown to be able to provide a good scenario to achieve this [11,12]. In A 4 models, the charged lepton mass matrix M l is diagonalized from left (rotation on left-handed charged leptons) by the characteristic matrix U l for A 4 symmetry model buildings [7], where ω = exp(i2π/3) and ω 2 = exp(i4π/3). U r is a unitary matrix, but does not play a role in determining V P M N S . We will not specify its form here. If neutrinos are Majorana particles, the most general mass matrix is of the form which can be diagonalized by The mixing matrix V P M N S is given by In the basis where charged lepton is diagonalized, the neutrino mass matrix is of the form given by eq.(6) with [12] a = 1 3 (w 1 + w 2 + w 3 + 2(x + y + z)) , If one imposes the GLS on the neutrino mass matrix, all parameters in the set P = (w i , x, y, z) are dictated to be real, and will predict [11,12] δ = ±π/2 and θ 23 = π/4. Therefore in A 4 model building for neutrino masses with δ = ±π/2 and θ 23 = π/4, it is essentially to make sure that U l is of the form given by eq.(8) and require the resulting mass matrix to satisfy GLS. Note that in this case since the parameters in set P are all real, the complexity of the mixing matrix is purely due to the appearance of ω and ω 2 . When the GLS is broken by allowing the parameters in P can be complex, there are more sources for CP violation and the model does not predict δ = ±π/2 and θ 23 = π/4 automatically. This points a way to modify the predictions to fit data should future experiments will find δ and θ 23 to be deviate significantly from −π/2 and π/4. We will study both cases with the parameters in set P to be real and complex in the rest of the paper.

II. TYPE-II SEESAW MODEL WITH A 4 SYMMETRY
We now construct a Type-II seesaw model [13] with A 4 family symmetry to realize the forms of mass matrices in eqs.(6) and (7). A different model based on Type-II seesaw with A 4 has been constructed to realized tribi-maximal neutrino mixing [14]. In our model, the left-handed lepton doublet l L and right-handed charged lepton singlet l R have the following standard SU(3) C × SU(2) L × U(1) Y gauge and A 4 family symmetry properties where the first three numbers in the first bracket indicate the SU(3) C , SU(2) L and U(1) Y transformation properties. The numbers in the second bracket indicate the A 4 representations.
To obtain desired mixing pattern, the Higgs sector is enlarged to have two types of Higgs doublets, φ and Φ, and two triplets, ∆ and χ for neutrino masses. They transform as The Lagrangian responsible for the lepton mass matrix is If the structure of the vacuum expectation value (vev) is of the form ∆ , one obtains the charged lepton and neutrino mass matrices M l and M ν as where If the vevs of < Φ 1,2,3 >= v Φ 1,2,3 are all equal to v Φ , the vev structure of the Higgs fields breaks A 4 , but left with a Z 3 residual symmetry generated by {I, c, a}. Here c, a are A 4 group elements defined in Ref. [15]. This will lead to the charged lepton mass matrix of the form where m e,µ,τ = √ 3y e,µ,τ v Φ . As long as Z 3 residual symmetry in the model is not broken, we have one of the key ingredients in realizing the form of charged lepton mass matrix in eq.(8). In the basis where the charged lepton mass matrix is diagonalized, the neutrino mass matrix will be the same form in eq.(6).
In the rest of this paper, we will study consequences related to the above mass matrix. We will study conditions on the model parameters imposed by GLS, and phenomenologically acceptable models can result in this class of models.
Before carrying out detailed analysis for neutrino mixing, we would like to point out that this model can easily accommodate data in the quark sector if one assigns the left-and right-handed quarks Q L , U R ad D R as A 4 singlet "1". In this case the Yukawa couplings for quark masses and their mixing are given by This will give, in general, arbitrary 3 × 3 up-and down-quark mass matrices M u,d after φ develops a non-zero vev v φ with M u,d = Y u,d v φ . These matrices will have no predictive power for quark masses and their mixing, but have no problem in accommodating experimental data.
If the parameters in the set P are all real, the resulting mass matrix is of the form given in eq. (7). Therefore this model predicts .
The above prediction can also be obtained by studying the mixing matrix V P M N S is eq.(10). For an arbitrary complex symmetric matrix M ν , the matrix V ν diagonalizes it, has the most general form Here we have absorbed a Majorana phase matrix diag(1, e iη 2 , e iη 3 ) on the right of the above equation into the eigen-masses m i e i2η i . If the parameter set in P are all real, the mass matrix M ν can be diagonalized by an orthogonal mixing matrix V ν = U(θ 12 ,θ 13 ,θ 23 ,δ = 0), and e iτ i are real. Depending on conventions, heres ij and c ij need not to be normalized all to be positive. We have the V P M N S given as the following The above matrix has the property that, |V P M N S,2i | = |V P M N S,3i |. Rewriting the above into the standard form of the mixing matrix, this leads to [9] θ 23 = π 4 , s 13 cos δ = 0.
Since experimentally s 13 = 0, then cos δ = 0 which leads to δ = ±π/2. The other angles can be determined by where The condition whether δ takes +π/2 or −π/2 is determined by using the Jarlskog parameter Note that ifδ = 0 and/or e iτ i are not real, in general |V P M N S,2i | is not the same as |V P M N S,3i |. θ 23 and δ will not necessarily be π/4 and ±π/2. One should be cautious about this. This uncertainty actually can be turned into a good use to provide a possible way to modify the predictions for θ 23 and δ. The value for θ 23 is determined by The angles θ 13 and θ 12 are determined by eq.(23), and the CP violating phase δ is given by eq.(24).
The above discussions provides a way of looking at how the reality of the parameters in the set P leads to δ = ±π/2 and θ 23 = π/4. It is essential to have the parameters in the set P to be all real. The complexity of the parameters can appear in the Yukawa couplings, in the vevs, and also in places where ω i appear in neutrino mass matrix. One needs to see with what conditions they can be made real.
To make the Yukawa couplings and scalar vevs real, one can require the model Lagrangian to satisfy a generalized CP symmetry under which and all other fields transform the same as those under the usual CP symmetry. Here the superscript CP in the above indicate that the fields are the usual CP transformed fields. The above transformation properties will transform relevant terms into their complex conjugate ones. Requiring the Lagrangian to be invariant under the above transformation dictates the Yukawa couplings to be real. The same requirement will dictates the scalar potential to forbid spontaneous CP violation and vevs to be real. One, however, notices that the parameters w 2,3 in eq.(16) are in general complex even if the Yukawa couplings and the vevs of the scalar fields are made real because the appearance of ω i . To make them real to satisfy GLS, it is therefore required that The above can be achieved by the absent of the scalar fields ∆ ′ , ′′ in the theory or Y ν v ∆ ′′ . We will consider examples for each case in the next section.
From the above discussions, we know that in general the mass matrices obtained can accommodate values different than π/4 and ±π/2 for θ 23 and δ respectively, we will not carry out a full numerical search analysis for parameter spaces, but to take some simple cases to show how the modifications arise when GLS is broken and how future experiments can test this class of models. These models, due to additional constraints, will have some additional predictions than the general one.
A. Model A: Z 2 residual symmetry for neutrino mass matrix If vev of χ 2 component of χ is non-zero, but the vevs of χ 1,3 are zero, the vev structure breaks A 4 down to a Z 2 generated by {1, r 2 }. Here r 2 is an A 4 group element defined in Ref. [15]. The charge lepton and neutrino mass matrices are given by The parameters in the set w i , and y are in general complex. If the fields, ∆ ′ and ∆ ′′ are absent from the model, one would have w 1 = w 2 = w 3 = Y 0 ν v 0 ∆ . The resulting mass matrices lead to the well known tribi-maximal mixing [17] which had been the focus for A 4 symmetry model buildings before [15,[18][19][20]. In this case, θ 13 = 0 and phase δ is zero. Since experimentally s 13 has been determined to be non-zero, this model is ruled out. Modifications have to be implemented [21][22][23] to accommodate data. A way out is to keep Y ′ ν v ∆ ′ = Y ′′ ν v ∆ ′′ in the model as discussed in the previous section.
We now summarize the main results follow Ref. [11] for this model. This also services as an outline how analysis can be carried out for relevant models. Letting M ν be diagonalized byṼ ν , one has V ν in this case can always be written in the following way with Majorana phases to be absorbed in to the neutrino masses m i ,Ṽ where V ρ is a diagonal matrix diag(1, 1, e iρ ) with tan ρ = Im(yw * 1 +y * w 3 )/Re(yw * 1 +y * w 3 ). s = sin θ and c = cos θ.
Expressing the mixing angle θ in terms of the model parameters, we obtain , we obtain the Majorana phases α i of m i in terms of the model parameters as One can always normalize the Majorana phase α 2 to be zero without of generality. In this basis, the phase of w 2 is also zero. Although the Majorana phases can be expressed in terms of the model parameters, since there are more number of parameters than the mixing angles and eigen-masses, there is no prediction for the Majorana phases. We will not discuss them in our later numerical analysis any more. The mixing matrix V P M N S can be, in general, written as From the above, one clearly sees that if sin ρ is not zero, |δ| and θ 23 deviate from π/2 and π/4, respectively. In the limit ρ goes to zero, however, the above recovers the results with real parameter set P with δ = ±π/2 and θ 23 = π/4. There are two interesting features for this model worth mentioning. One of is that |V e2 | to be 1/ √ 3 which agree with date. s 12 is always larger or equal to 1/ √ 3 which is a decisive test for this model. Another is that although the Dirac phase δ depends on the phase ρ, the Jarlskog parameter J which is independent of ρ given by J = −(c 2 − s 2 )/6 √ 3. This implies that CP violation related to neutrino oscillation is still purely due to intrinsic CP violation. This model can be made in agreement with data at 1σ level.
If ρ = 0 and c = s = 1/ √ 2, the mixing pattern is the tribi-maximal. However, if ρ is not zero, even if c = s = 1/ √ 2, s 13 can be non-zero,s 12 and s 23 are also modified from their tribi-maximal values J is exactly zero which implies sin δ = 0.
B. Model B: mass matrix with w 1,2,3 = 0 Without ∆ 0, ′ , ′′ , w 1,2,3 = 0. In this case all the three vevs of χ should not be equal in order to fit data on the neutrino mass-squared differences ∆m 2 21 = |m 2 | 2 − |m 1 | 2 and ∆m 2 31 = |m 3 | 2 − |m 1 | 2 . The Z 2 residual symmetry mentioned previously is also broken in this case. If CP is spontaneously broken in the Higgs potential, v χ i may be complex too. Therefore the parameters in the set P : {x, y, z} = {|x|e iσx , |y|e iσy , |z|e iσz } are in general complex. One can always choose, without loss of generality, to rewrite the neutrino mass matrix as where σ 1 = (σ z − σ x − σ y )/2, σ 2 = σ z − σ y and σ 3 = σ x − σ y . σ 1 can be absorbed into lepton fields. We will neglect it in the following analysis. Let us discuss now how the model parameters can be determined by data. First of all sinceM ν is a real matrix, the eigen-valuesm i are all real andm 1 +m 2 +m 3 = 0. The absolute value of neutrino mass |m i | = |m i | can be solved as functions of ∆m 2 21 and ∆m 2 31 . For the values allowed by experimental data, the solutions are given by up to a overall sigñ One can easily check that multiplying a "−" to the above one obtains another solution. The overall sign can be absorbed into lepton field phases redefinition. We will use the above normalization for the overall sign. In the numerical study later, we will show that only inverted hierarchy is allowed. Therefore, m 1,3 > 0 and m 2 < 0. Since the eigen-masses are all real, there is no Majorana phases in this model. SinceM ν is real, it can be diagonalized by a orthogonal matrix. Assuming U(θ 13 ,θ 13 ,θ 23 ,δ = 0) diagonalizesM ν , one can express the fact that the "11"' and "22" entries are zero as wheret ij =s ij /c ij . From the above two equations, one can expresss 12 ands 23 as functions ofs 13 andm i . Sincẽ m i are known, the elements in V P M N S = U † l V σ V ν can be expressed as functions of σ 2 , σ 3 ands 13 . Using the relations between the elements V ij of V P M N S with s ij and δ in eqs. (23) and (24), one can vary σ 2 , σ 3 ands 13 to see if the resulting values for s 12,13,23 are in the allowed values, and obtain δ by eq.(24).
Similar analysis had been used to rule out [24] the simple version of Zee model where the mass matrix for neutrinos isM ν in the diagonalized basis of the charged leptons. Here the appearance of U l may save the model. We find that, unfortunately, that with real parameters in P which implies σ 2,3 to be zero, δ and θ 23 are predicted to be −π/2 and π/4, but there is no solutions within the allowed ranges which can predict s 12 to be consistent with data. With complex parameters in P , the values for s 12,13,23 can be in agreement with data, it is, however, not possible to get δ to be close to −π/2. More numerical details will be provided in the next section.

IV. NUMERAL ANALYSIS
In this section, we compare experimental data with the model predictions for the mixing angles and CP violating phase. There are several global fits of neutrino data [3,4]. The latest fit gives the central values, 1σ errors and the 2σ ranges as the following [ Here NH and IH indicate neutrino mass hierarchy patterns of normal hierarchy and inverted hierarchy, respectively. We will use the above data for comparison.

A. Some general comments
The model discussed achieved a natural way to have the CP violating angle δ to be ±π/2 and θ 23 to be π/4. The condition is to have all the parameters in the set P to be real. This requirement can be taken as a consequence of having CP violation only caused by intrinsic source of CP violation. The condition for choosing δ = −π/2 can be determined using eq.(24) which requires Im(V 11 V * 12 V * 21 V 22 ) to be negative. The value −π/2 predicted in the model is in agreement with IH within 1σ range. Although for NH case δ is outside of 1σ range, there is no problem with 2σ range. For s 23 , the model predicts s 2 23 = 0.5. This value is outside of 1σ range for both the NH and IH cases. However, they are, again, in agreement with data within 2σ. Other parameters, ∆m 2 ij , s 13 and s 12 can be easily fitted to within 1σ level of current experimental data.
As the general model is expected to be able to fit data, it may be more instructive to analyze some simplified versions than just providing with numbers. We provide more details of Model A and Model B discussed earlier next to see how additional assumptions restrict the level of model agreement with data.

B. Model A predictions
With real values for w i and y in Model A, by adjusting the values, w i , and y both NH and IH mass patterns can be obtained. The predictions for δ and θ 23 are ±π/2 and θ 23 = π/4, just like the general case discussed in the previous section. Since δ should be close to −π/2, one should take the parameter space so that c 2 > s 2 . In the model s 13 = (1 − 2cs) 1/2 / √ 3 is not predicted. But one can use information from s 13 to fix cs = 0.497 ± 0.018 to predict s 2 12 = 0.334 ± 0.004 for both NH and IH cases. This is in agreement with data within 1σ. Note that V 2 e2 = (s 12 c 13 ) 2 = 1/3. It agrees with data within 1σ.
It is remarkable that neutrino mixing matrix in this model with just one free parameter can be in reasonable agreement with data. This may be a hint that it is the form for mixing matrix, at least as the lowest order approximation, that a underlying theory is producing. One should take this mass matrix seriously in theoretical model buildings.
If the parameters in the set P are complex, and therefore a new phase ρ appears in the model. In this case, the new parameter ρ can be used to improve agreement of the model with data. In both NH and IH cases, δ and s 23 can be brought into agreement with data at 1σ level. As an example, we take the largest value of cs so that s 13 takes its lower 1σ allowed value, and then varying cos ρ to obtain the upper 1σ allowed value. This fixes cs and cos ρ to be 0.468 and 0.992, respectively. With these values, s 23 and δ are determined to: 0.534 and 1.426π, respectively. These values are in agreement with data at 1σ level.
For the case with c = s, the model is more restrictive. In this case sin δ = 0. With more precise data on the CP violating angle δ, this may rule out this simple case with high confidence level. If ρ = 0, the model is already ruled out at high precision from s 13 measurement. However, with a non-zero ρ, the mixing angles can still be made to in agreement at 2σ level. In Fig. 1 we show s 12 , s 23 , s 13 as functions of ρ. When chose cosρ = 0.93, we can get s 12 = 0.58, s 23 = 0.78, s 13 = 0.15 which agree with the experimental data within 2σ range. More precise experimental data are required to distinguish the model with complex model parameters from that with the real parameters and other models, or to rule out the above simples completely.

C. Model B predictions
With real parameters in P in this model, the predictions, δ and θ 23 are the same as the general case. Using eq.(39), one can chooses 13 to be the parameter to fit all data. We find that no solutions can simultaneously bring s 13 and s 12 to be in 2σ allowed region compared with data. In Fig. 2 we show s 13 , s 12 as functions ofs 13 . Here, when s 13 agree with the data in 2σ range, the allowed region fors 13   With complex parameters in P , there are solutions ofs 13 and σ i so that to make s 23 , s 13 and s 12 to be consistent with data at 1σ level. However this requires one of | sin σ i | to be away from zero which alter the prediction for δ significantly away from −π/2, although θ 23 is still close to π/4. In Fig. 3, we scan the parameter space −1 ≤s 13 ≤ 1, 0 ≤ σ 2 ≤ 2π, 0 ≤ σ 3 ≤ 2π, and obtained the corresponding values of s 12 , s 23 , s 13 that agree with data within 1σ and 2σ region respectively. And in 1σ range, the CP phase will be constrained in sinδ ∼ (−0.5, 0.5), but in 2σ region, CP phase is free and can be in the range of 0 to 2π. For example withs 13 = 0.57, σ 2 = 7π/5, σ 3 = 26π/25, one can obtain that s 13 = 0.15, s 23 = 0.74, s 12 = 0.57. These mixing angles are within the 1σ allowed region, but it will have a CP phase δ = 1.06π significantly away from 3π/2. A precise determination of the CP violating phase is needed to rule out the above model.

V. CONCLUSIONS
We have constructed theoretical models for neutrino mass matrix in Type-II seesaw with A 4 family symmetry. The models we constructed naturally predict that the CP violating phase δ is equal to ±π/2 and at the same time the mixing angle θ 23 is π/4. The reality of the parameters can be achieved by imposing a generalized CP symmetry in the Grimus-Lavoura symmetry limit. When the generalized CP symmetry is explicitly broken, the Yukawa couplings can be complex, the model predictions for δ and θ 23 can be significantly modified. Two simple scenarios, Model A and Model B, are analyzed in detail to show how the modifications arise and how future experimental data can test this class of models.
The Model A has a characteristic prediction that |V e2 | = 1/ √ 3. This model can accommodate experimental data at 1σ level. It can be taken as the lowest order neutrino mass matrix for future theoretical model buildings. When Yukawa couplings are complex, the CP violating phase δ and the mixing angle θ 23 can be away from −π/2 and π/4. The crucial test for this model is to measure whether |V e2 | = 1/ √ 3 holds to high precision. In Model B, the diagonal entries of neutrino mass matrix are all zero in the weak interaction basis. This implies that the neutrino masses can be determined by the known neutrino mass-squared differences. We find that this model can only accommodate inverted neutrino mass hierarchy. The mixing angles s 12 , s 13 and s 23 cannot simultaneously be in agreement with data at 2σ level if the Yukawa couplings are all real. With complex Yukawa couplings, the mixing angles can be brought into agreement with data at 1σ level, but the CP violating angle δ will be significantly away from −π/2. This provides a crucial test for this model.
At present, experimental data on neutrino mixing seem to hint that δ and θ 23 to be close to −π/2 and π/4. Theoretical models which can naturally achieve such predictions are interesting to study. The models we have constructed have many novel properties and can be tested. We will have to wait future experimental data to tell us more whether this class of neutrino mass matrix will survive.