Majorana CP Violation in Approximately  Symmetric Models with det(M)=0
1 Introduction
Neutrinos are oscillating and mixed with each other among three flavor neutrinos. Such oscillations have been confirmed to occur for the atmospheric neutrinos [1], the solar neutrinos [2, 3], the reactor neutrinos [4] and the accelerator neutrinos [5]. Three massive neutrinos have masses measured as mass squared differences defined by and . Three flavor neutrinos are mixed into three massive neutrinos during their flight and the mixing can be described by the PontecorvoMakiNakagawaSakata (PMNS) mixing matrix [6] parameterized by three mixing angles , one Dirac CP violating phase and three Majorana phase [7], where Majorana CP violating phases are given by two combinations of .
It is CP property of neutrinos that has currently received much attention since the similar CP property of quarks has been observed and successfully described by the KobayashiMaskawa mixing matrix [8]. If neutrinos exhibit CP violation, there is a new seed to produce the baryon number in the Universe by the FukugidaYanagida mechanism of the leptogenesis [9], which favors the seesaw mechanism [10] of creating tiny neutrino masses. However, there is no direct linkage between CP violation of three flavor neutrinos and that of the leptogenesis since the CP violating phases are associated with heavy neutrinos but not with three flavor neutrinos. If the number of the heavy neutrinos is two, there is onetoone correspondence between the CP violating phases of flavor neutrinos and that of the seesaw mechanism. The model with two heavy neutrinos is called minimal seesaw model [11]. Even if the minimal seesaw model is adopted, predictions of the leptonic CP violation depend on the choice of various parameters including Dirac neutrino mass terms. It seems of great significance to make predictions independent of the specific parameter choice. The general feature of the minimal seesaw model is that it satisfies , where represents a flavor neutrino mass matrix. Therefore, we choose this condition as our standing point to investigate effects of leptonic CP violations as general as possible.
To discuss the leptonic CP violation starting from a given phase structure of , we have to clarify how phases of flavor neutrino masses affect the leptonic CP phases. To do so, we have to mathematically consider 6 phase parameters in the PMNS mixing matrix to cover a general phase structure of . It should be noted that conventional studies utilizing the standard version of the PMNS matrix given by the Particle Data Group (PDG) [12] and do not provide a clue to see direct effects from phases of the flavor neutrino masses [13]. In our study, is parameterized by with
(1.1)  
where and (=1,2,3). There is the similar phase to and , say , which can be associated with the  mixing. However, we have confirmed that the phase of the  mixing should be in place of to consistently describe the neutrino mixings [14]. Namely, Eq.(1.1) with included is converted into with , , , and as obvious replacements. Physically, among , , and two phases are redundant. By defining (as well as ) and
(1.2) 
we reach consisting of
(1.3) 
Furthermore, one Majorana phase is also redundant and the CP violating Majorana phases are given by two combinations of the Majorana phases such as (). For the reader’s convenience, we show three typical forms of in Appendix A.
The neutrino mixing angles and mass squared differences have been measured by recent neutrino oscillation experiments. The current data of the mixing angles and mass squared differences are shown as [15]:
(1.4) 
The gross property of the experimental data indicating the almost maximal atmospheric neutrino mixing and the small 13 neutrino mixing can be understood as a result of the  symmetry [16] imposed on neutrino interactions, which gives and . However, there is no Dirac CP violation. If Dirac CP violation is observed in future neutrino experiments [17], we have to include tiny violation of the  symmetry.^{‡)}^{‡)}‡)The  symmetry breaking should be present because the charged leptons placed into doublets together with the flavor neutrinos violate the  symmetry. If the  symmetry breaking is included, there are two categories of textures respectively referred to as (C1) and (C2) [18]. In the category (C1), we have () and while in the category (C2), we have and . In the category (C2), the  symmetric limit is signaled by instead of . The phenomenologically consistent value of is realized by the form of , where represents the  symmetry breaking parameter and another small parameter denoted by of is required.
In this article, we discuss CP property of approximately  symmetric models satisfying , whose theoretical foundation is supplied by the minimal seesaw model with two heavy right handed neutrinos. We estimate sizes of CP violating phases by using the general phase structure of neutrino mass matrix and by focusing on the rephasing invariance, whose existence in our formalism is discussed in Appendix A . Some of results of the category (C1) are shared by our previous work [19]. All CP phases are expressed in terms of flavor neutrino masses so that one can understand that how phases of flavor neutrino masses induce CP violating phases in .
In the next section, we define the  symmetry and explain two categories (C1) and (C2). In Sec.3, we present various formulas to extract general property of the observed neutrino mixings and discuss how the condition of det()=0 gives a massless neutrino to exclude the case of . Detailed discussions to see the appearance of one massless neutrino are given by Appendix B. In Sec.4, we include effects of the  symmetry breaking to see neutrinos in the categories (C1) and (C2), which are used to construct neutrino mass textures. In Sec.5, we argue how mass hierarchies are realized and show seven viable textures, where we estimate CP violating phases in each texture. The last section is devoted to summary and discussions.
2  symmetry
The  symmetry is the symmetry due to the invariance of the lagrangian, especially for the flavor neutrino mass term , associated with the interchange of . We define the interchange as follows:
(2.1) 
where will take care of the sign of as parameterized by Eq.(1.1). The phase turns out to be of the  symmetry breaking type. Our mass term is labeled by
(2.2) 
which is divided into and (+):
(2.3) 
Under the interchange Eq.(2.1), is kept intact. From this result, the superscripts and of are, respectively, so chosen to stand for the  symmetry preserving and breaking terms.
We find that is determined by (0, , ) as one of the eigenvectors associated with if it is assigned to . One may also assign it to giving , even to giving . Namely , respectively, takes the form of
(2.4) 
There are in principle three possibilities for  symmetric textures. However, for , after the  symmetry is broken, it can be shown that (1) but , which contradicts the data Eq.(1.4). As a plausible choice, we obtain [18]
(2.5) 
as a category (C1) or
(2.6) 
as a category (C2). The phase is determined as =arg() to be shown in Eq.(3.8), where stands for an  element of (=) defined in Eq.(2.2). Both categories give no Dirac CP violation signaled by or by . In other words, the Jarlskog invariant [20] vanishes. In models without leptonic CP violation, both cases can produce experimentally allowed results [18], where the  symmetry breaking is a must for the category (C2).
3 Various Relations of Masses and Mixings
To extract general property inherent to the observed neutrino mixings, we first derive various formulas expressed in terms of flavor neutrino masses to evaluate neutrino masses, mixing angles and phases.
3.1 Useful Formulas
To get the Dirac CP phase as well as and , it is convenient to use parameterized by
(3.1) 
where Majorana phases are removed. From
(3.2) 
we obtain that
(3.3)  
(3.4)  
(3.5)  
(3.6)  
(3.7) 
Because and are real numbers, the phases and can be determined by
(3.8) 
The remaining phase is determined from Eq.(3.4). It is further obtained that the size of should be suppressed to realize the hierarchy of . where
(3.9) 
To calculate the Majorana phases, we instead diagonalize :
(3.10) 
and find that
(3.11)  
(3.12)  
(3.13)  
(3.14) 
If is suppressed, to give a sizable . The Majorana phases become the similar order for the inverted mass hierarchy with . In this case, for det()=0 giving , Majorana CP violation is generically small.
3.2 det(M)=0
Let us next see how the condition of provides a massless neutrino when our formulas of are used. Since the violation of the  symmetry is tiny, it does not significantly affect the sizes of the neutrino masses evaluated in the  symmetric limit although it affects the size of for the category (C2). The obtained results are to be used to construct textures either with or but not with .
If we parameterize as follows:
(3.15) 
is translated into:
(3.16) 
where . The factor may not be needed in Eq.(3.16); however, it intends to take care of from in the  symmetric limit and it is merely our matter of convention. In the  symmetric limit, Eq.(3.16) turns out to be
(3.17) 
for, and
(3.18) 
for . One can also confirm that the condition of evaluated up to the first order of the  symmetry breaking coincides with that of in the  symmetric limit.
The appearance of a massless neutrino is described in Appendix B using Eq.(3.12). The results are summarized as follows:

from Eq.(3.17),
for the category (C1), where and in the  symmetric limit, and

from Eq.(3.17)
for the category (C2), where and in the  symmetric limit. In any cases, the massless should not be realized by textures.
4 Effect of  Symmetry Breaking
To specify phase structure of , let us first count phases present in . There are six complex numbers in . Since three phases are removed by the rephasing, among the remaining three phases, one phase can be determined by . We are left with two phases, which are taken to be the phases associated with and . For the present discussions, these phases are denoted by associated with and by associated with in place of and for the sake of convenience. Any other choices give the same results of CP violation because of the rephasing invariance as shown in Appendix A. Thus, our results do not depend on our specific choice of phases in and will cover leptonic CP properties in models with , where the charged lepton masses are necessarily taken to diagonal [21].
4.1 Parameterization of  Symmetry Breaking
To describe the  symmetry breaking flavor neutrino masses, we parameterize
(4.1) 
for Eq.(2.3) and
(4.2) 
where . We show results valid up to the terms of .^{§)}^{§)}§)It should be noted that the smallness of is naturally in the category (C2) but the smallness of is also implicitly assumed to be consistent with the experimental observation. It is realized by the smallness of contained in (See Eq.(4.5) for ). To do so, we parameterize as follows:
(4.3) 
where is responsible for the effect of . The parameter takes care of the difference of the category: for the category (C1) and for the category (C2).
4.2 Masses, Mixing Angles and Phases
Before we give explicit textures, we calculate masses, mixing angles and phases in terms of the mass parameters of Eq.(4.1) from relations found in the previous subsection for each category. Explicit forms of textures can readily be obtained once we give mass parameters in Eq.(4.1), which are taken to realize the normal mass hierarchy or the inverted mass hierarchy.
4.2.1 Category (C1)
Mixing angles and Dirac phases can be estimated in terms of Eq.(4.4) as
(4.6) 
where
(4.7) 
leading to
(4.8) 
from . Since involves via and in its righthand side, one can further solve to give
(4.9) 
where and are given by Eq.(4.4), which gives . Neutrino masses and Majorana phases are estimated to be:
(4.10) 
for the normal mass hierarchy, and