Effects on sin\theta_{12} from perturbation of the neutrino mixing matrix with the partially degenerated neutrino masses

We consider a situation where the leading-order neutrino mass matrix is derived by a theoretical ansatz and reproduces the experimental data well, but not completely. Then, the next stage is to try to fully reproduce the data by adding small perturbation terms. In this paper, we obtain the analytical method to diagonalize the perturbed mass matrix and find a consistency condition that parameters should satisfy not to change \sin\theta_{12} much. This condition could cause parameter tuning and plays a crucial role in relating the added perturbation terms with the prediction analytically, in particular, for the case of the partially quasi-degenerated neutrino masses (m_2 \simeq m_1) where neutrinoless double beta decays would be observed in the phase-II experiments.


Introduction
Various types of the neutrinoless double beta decay (0νββ) experiments have been working, and the phase-II experiments are planned; see Refs. [1,2], [3,4,5], and [6,7] for the recent reviews, studies with cosmological observations, and previous works, respectively. In these experiments, the expected sensitivity to the effective neutrino mass, m ν , would hopefully reach to 0.02 eV. As discussed by many authors, if the observed effective mass m ν is in regions much larger than ∆m 2 a ≃ 0.049 eV, the possible mass pattern is the quasi-degenerate (QD) one (see Fig. 1). If m ν is greater than or equal to 0.049 eV, the inverted hierarchy (IH) with the constructive interference of the Majorana CP phases between m 2 and m 1 would be favored. For the IH case with the destructive interface, m ν is greater than or equal to 0.014 eV. In the case of the normal hierarchy (NH), with the sensitivity of m ν > 0.02 eV, one could explore the partially-quasi-degenerated mass regions, in which m 2 ≃ m 1 ; especially, most of the constructive interference regions would be covered. Thus, these parameter regions are expected to be important in the coming years.
Let us suppose that the leading-order neutrino mass matrix M 0 is derived theoretically with use of some symmetry and that its diagonalizing matrix V 0 , which is defined by reproduces the experimental data of the mixing angles well, but not preciously. In particular, (V 0 ) 12 is assumed to be very close to its experimental value. Here, m 0 i are taken to be real and positive, and β 0 and α 0 are their CP phases. In order to fill the gap between V 0 and the experimental data, we add three small complex parameters: 1 Figure 1: The effective mass, mν , of the 0νββ as functions of the lightest neutrino masses, m 1 (m 3 ) for the NH (IH) case; all the CP phases are varied from 0 to 2π; the gray (red) region is allowed by the 3σ constraints of the oscillation parameters [8]  In model-building, we put some restriction on the parameters ǫ i and obtain prediction. Our question is to see analytically the relation between the restriction and the prediction. For this, we have to diagonalize the neutrino mass matrix analytically as precisely as possible and then expand the exact result in terms of small parameters. We develop a powerful method to achieve this and see the relation between the restriction and the prediction. In the course of this, we find a constraint which parameters should satisfy in order not to change sin θ 12 much. This is the condition for the model to be consistent. In view of the observability in the future 0νββ experiments, we are mainly interested in the partially-quasi-degenerated mass regions and pay special attention to three cases: the NH with the constructive interference case and the IH 3 with the constructive and destructive interference cases. Nevertheless, we sometimes consider the other cases for the sake of completeness. In Sect. 2, we review the behavior of the effective mass of the 0νββ with respect to p = m 2 /m 3 and the Majorana phases [12,13] for the purpose of the following sections. In Sect. 3, the diagonalization of a symmetric matrix with small perturbation terms is developed, and then the consistency condition which guarantees that sin θ 12 does not change much is derived in Sect. 4. In Sect. 5, the developed method is applied to the case of the tri-bi-maximal mixing, and the relations between the restriction of parameters and the prediction are given for various models in Sect. 6. The concluding remarks are given in Sect. 7.

Behavior of effective mass of 0νββ
In the introduction, we argued that our main interests are the NH case for the regions of m 2 ≃ m 1 with the constructive interference, and the IH cases with both the constructive and destructive interferences. We here summarize the behavior of m ν for these cases. In the followings, we use the convention that the mass parameter m i are real and positive and that m 2 and m 1 are accompanied by the Majorana phases β and α, respectively. These Majorana phases appear in the mixing matrix as the phase matrix • The NH case for the regions of m 2 ≃ m 1 with the constructive interference β ≃ α In this case, p < 1 and neutrino masses are expressed as The effective mass is written by where s ij (c ij ) stands for sin θ ij (cos θ ij ), and we have used s 13 ≪ 1. For m ν > 0.02eV, one finds p > 0.4.
• The IH case In this case, p > 1 and On one hand, the effective mass for the destructive interference case is m ν ≃ m 2 |c 13 cos 2θ 12 | ≃ | cos 2θ 12 | for the 3σ upper bound sin 2 θ 12 < 0.359 [8]. On the other hand, the constructive interference case is

Diagonalization of symmetric matrix with small perturbation terms
We supplement the leading-order neutrino mass matrix Eq. (1) by the small perturbation terms in Eq.
(2) and define the full mass matrix as where 3 and the overall factor µ stands for the heaviest mass among m 0 i : µ = m 0 3 (m 0 2 ) for the NH (IH) case. We emphasize that this is the most-general complex symmetric matrix in the sense of the number of parameters. Throughout this paper, we choose a basis in which the charged lepton mass matrix is diagonal and k 3 is real and positive.
We first make M block diagonalized by the unitary matrix V 1 : where and This V 1 mainly affects sin θ 13 and sin θ 23 4 , and ǫ 3 and ǫ 2 are of the orders of f and g, respectively, as we shall see later. After this transformation, we find where and m 3 = µ|L|. We require the element N vanishes, which leads to This identity relates ǫ i 's with f and g, but we postpone showing their expressions till Eq. (24). The 2 × 2 matrix K is parametrized as which can be expressed explicitly in terms of ǫ i 's, but it is also postponed to Eq. (26). As we shall see later, |b| ≫ |c|. Next, we diagonalize the matrix K by the unitary matrix V 2 : where C = cos Θ and S = sin Θ, and V 2 affects θ 12 . The important point is that S will be much smaller than f and g, because we assume that (V 0 ) 12 is very close to the experimental data. The angle Θ and the phase κ are given by respectively. The eigenvalues are found to be where µ is the overall factor defined in Eq. (9), and m 1,2 are the physical neutrino masses, which are real and positive. From them, the mass splitting between m 2 and m 1 is written as and we find The neutrino mixing matrix is obtained by (V 0 V 1 V 2 ) aside from phases of neutrino masses, which are related to the Majorana phases.
Up to now, the analysis is exact. In what follows, we exploit the fact that ǫ 3 and ǫ 2 (thus, f and g) are small and that ǫ 1 is much smaller than them: as we shall show later, ǫ 1 should be of the order of ǫ 2 2,3 or much smaller than it. We hereafter omit terms which are higher than f 2 , g 2 and terms proportional to ǫ 2 f and ǫ 2 g. In this case, Eq. (16) is reduces to be or and the parameters a, b, and c included in K are expressed as Now, we can compute in a good approximation the neutrino mixing matrix V = (V 0 V 1 V 2 ), once V 0 is given.

Consistency conditions
One may think that the mixing angles are only moderately corrected since ǫ's are assumed to be small. However, S is not necessarily small; rather, it could take an unrealistically-large value. This is because the denominator of Eq. (22) is preciously measured and is very small. In order for the full mixing matrix V being consistent with the experimental data, therefore, one needs to somehow make the numerator sufficiently small, which leads to We hereafter refer to this requirement as consistency condition. In the followings, we further examine it by categorizing the neutrino mass spectrum into three types.
1. The NH case in the regions of m 2 ≫ m 1 .
In the case of NH, µ = m 0 3 and With Eq. (26), the left-hand side of Eq. (27) is written by Note that m 0 i are taken to be real and positive, and g is given in Eq. (24). Since m 2 ≃ m 0 2 and m 2 ≫ m 1 , the term proportional to m 0 1 may be dropped in comparison with that of m 0 2 . By using the approximations m 0 3 ≃ m 3 ≃ ∆m 2 a and p 0 = m 0 2 /m 0 3 ≃ p ≃ ∆m 2 s /∆m 2 a , we find 2. The NH case in the regions of m 2 ≃ m 1 (p > 0.4). This case occurs when the neutrinoless double beta decay is observed in the phase-II experiments. By taking the limit of m i = m 0 i and m 0 2 = m 0 1 , the consistency condition can be rewritten as Note that β 0 ≃ β and α 0 ≃ α.
Since m 2 is always quasi-degenerated with m 1 , the consistent condition turns out In all the cases, the key ingredient is ǫ 1 − ǫ 3 g, and the consistency conditions force ǫ 1 to be of the order of ǫ 2 2,3 . In other words, one needs to tune ǫ 1 to cancel out ǫ 3 g. As we shall demonstrate in the next section, this causes unnatural parameter-tuning in some cases.

Tri-bi-maximal mixing case
We here choose the Tri-Bi-Maximal (TBM) mixing [14,15,16] In this case, the full mixing matrix after perturbation is obtained as up to the first order of f , g, and S. The mixing angles are derived as and where κ is defined in Eq. (19). We have taken into account the second order terms of f and g for sin 2 θ 12 as they could be the main contributor depending on the sizes of cos κ and S. Note that the orders of |f | and |g| are constrained by sin θ 13 and sin θ 23 , and their contributions to sin 2 θ 12 are at most ±0.01; in contrast, they are crucial when evaluating S as we outlined in Sect. 3. According to the latest global analysis by Capozzi et. al. [8], the allowed 2σ (3σ) range is 0.275(0.259) ≤ sin 2 θ 12 ≤ 0.342(0.359), which places The angle S ≃ Θ is much smaller than the first order term as long as cos κ is not very small. Even in the case cos κ = 0, S is the first order term.
In below, we examine the behavior of cos κ for the three cases.
1. The NH case in the regions of m 2 ≫ m 1 . We find a * c + bc * ≃ p 0 e iβ0 (ǫ 1 − ǫ 3 g) * , so that It may be worthwhile to note that β 0 is almost equal to the Majorana CP violating phase β because the phase of V 12 is suppressed and phases of V 23 and V 33 are absorbed by charged lepton fields.
2. The NH case in the regions of m 2 ≃ m 1 .
(41) 7 Namely, the former happens in the case of the constructing interference of the Majorana phases, while the latter is the case of the destructive interference. It should be noted that for cos κ = −1, the correction decreases sin θ 12 because we choose S ≥ 0, while for cos κ = 1 and cos κ = 0, the correction increases it. The present tendency seems to disfavor the cos κ = 1 and cos κ = 0 cases.

The IH case.
In this case, we find and thus Note that α 0 and β 0 are not necessarily equal to the physical Majorana phases when m 3 ≃ 0, but α 0 − β 0 ≃ α − β still holds 5 . Therefore, like the previous case, cos κ ≃ ±1 and cos κ ≃ 0 occur in the constructive and destructive interference cases, respectively.

Parameter tuning
Let us roughly estimate how strong the parameter-tuning required by the consistency condition is. Taking the limits of cos κ = −1 and cos κ = 0, we place | sin 2 θ 12 − 1/3| ≤ 0.025. This number corresponds to the best-fit-value and 3σ-upper-bound [8] for cos κ = −1 and 0, giving rise to Θ ≤ 0.027 and 0.28, respectively. Also, we will use ∆m 2 s /∆m 2 a = 0.031 and ignore the corrections from the second order terms of f and g.
1. The NH case in the regions of m 2 ≫ m 1 . The consistency condition is given in Eq. (30). For cos κ = −1, we find For cos κ = 0, the parameter-tuning is not so serious. We have substituted |ǫ 3 g| ≃ |f g| = 0.04 in view of sin 2 θ best 13 ≃ 0.023 [8]. 2. The NH case in the regions of m 2 ≃ m 1 .
As demonstrated above, from a few % to several tens of % tuning is required between ǫ 1 and ǫ 3 g. In particular, somewhat strong parameter-tuning may be necessary in the case of m 2 ≃ m 1 with β ≃ α.

Validity of consistency conditions
We numerically diagonalize the mass matrix and check the validity of the consistency conditions Eqs. (30), (31) and (33). In the numerical calculations, we place the 1σ error bounds for ∆m 2 s , ∆m 2 a , sin 2 θ 13 , and sin 2 θ 23 from Ref.
In Figs. 2 and 3, we plot sin 2 θ 12 as a function of the left-hand side of the consistency condition for Eqs. (30) and (31). The figures for Eq. (33) are almost the same as Fig. 3. In Fig. 2, m 1 = 0 and cos κ = ±1 are assumed. The left and right panels in Fig. 3 are the cases of the constructive interference (cos κ = ±1) and destructive interference (cos κ = 0), respectively, for p = 0.4 − 0.8. All the CP phases are varied from 0 to 2π, and |ǫ 2 | and |ǫ 3 | run from 0.00 to 0.25. From the figures, one can observe a trend that sin 2 θ 12 approaches to its TBM value as the consistency conditions are satisfied. In the left panel of Fig. 3, however, sin 2 θ 12 departs from the TBM value even if the x-axis is zero. This is due to the failure of the approximations made above Eq. (23), and this indicates that one needs to take into account next higher-order terms and tune ǫ 1 − ǫ 3 g to cancel out them. The resulting condition would be very complex and require much more delicate parameter-tuning. Hence, we do not go into its detail here. In the next section, we shall invent several models where the consistency conditions Eqs. (31) and (33)

Applications to models
As we demonstrated in the previous section, the consistency conditions could be satisfied by tuning ǫ 1 . However, it may be difficult to explain such parameter-tuning by model-building. Furthermore, in some cases, the consistency conditions fail to keep sin θ 12 within experimentally-realistic ranges. In this section, we consider two other possibilities by postulating ǫ 1 = 0: (1) adjusting either |ǫ 2 | or |ǫ 3 | to be very small, and (2) adjusting CP phases. In the models proposed below, the consistency conditions work very well. Moreover, they seem attractive from model-building and/or phenomenological points of view. For definition, we again employ the TBM mixing as V 0 .
In this case, the mixing matrix turns out to be the so-called tri-maximal mixing [17]: Its mixing properties have been extensively studied by many authors, so that we refrain from going into details. See, for instance, Refs. [18,19,20,21] for the behavior of sin 2 θ 12 and the others. Nevertheless, several comments are in order. (1) The higher-order term of f included in Eq. (36) slightly increases sin 2 θ 12 ; thus sin 2 θ 12 > 1/3 is predicted.
It may be interesting to note that the corresponding mass matrix preserves a Z 2 symmetry even after adding the perturbation terms. It is well known that the TBM mixing can be derived from the mass matrix invariant under the following Z 2 symmetries [22,23,24] (see also Ref. [25]): in the flavor basis. In the case of |ǫ 1 | = |ǫ 2 | = 0, G TBM 2 remains unbroken. This often happens in a class of the A 4 flavor model [26,27,28] because A 4 does not include G TBM 1 .
It should also be noted that the difficulty to keep θ 12 around the TBM value, while reproducing a large θ 13 , in the A 4 flavor model was pointed out in Refs. [29,30,31]. They arrived at the same solution, ǫ 1 = ǫ 2 = 0, and Eq. (51).
As is the case of the tri-maximal mixing, the mass matrix preserves G TBM 1 G TBM 2 in the flavor basis.
In this case, Eq. (56) provides us with