Evaluation of the Majorana Phases of a General Majorana Neutrino Mass Matrix: Testability of hierarchical Flavour Models

We evaluate the Majorana phases for a general $3\times3$ complex symmetric neutrino mass matrix on the basis of Mohapatra-Rodejohann's phase convention using the three rephasing invariant quantities $I_{12}$, $I_{13}$ and $I_{23}$ proposed by Sarkar and Singh. We find them interesting as they allow us to evaluate each Majorana phase in a model independent way even if one eigenvalue is zero. Utilizing the solution of a general complex symmetric mass matrix for eigenvalues and mixing angles we determine the Majorana phases for both the hierarchies, normal and inverted, taking into account the constraints from neutrino oscillation global fit data as well as bound on the sum of the three light neutrino masses ($\Sigma_im_i$) and the neutrinoless double beta decay ($\beta\beta_{0\nu}$) parameter $|m_{11}|$. This methodology of finding the Majorana phases is applied thereafter in some predictive models for both the hierarchical cases (normal and inverted) to evaluate the corresponding Majorana phases and it is shown that all the sub cases presented in inverted hierarchy section can be realized in a model with texture zeros and scaling ansatz within the framework of inverse seesaw although one of the sub case following the normal hierarchy is yet to be established. Except the case of quasi degenerate neutrinos, the methodology obtained in this work is able to evaluate the corresponding Majorana phases, given any model of neutrino masses.


Introduction
Apart from hierarchical structure of massive neutrinos a fundamental qualitative nature of these elusive particles whether they are Dirac or Majorana type is yet unknown. Neutrinoless double beta decay (ββ 0ν ) mode [1]- [8] is able to discriminate between the two different types. Positive evidence of the above experimental search will be able to determine the Majorana nature of neutrinos assuming the above decay is mediated due to light neutrino. Several ββ 0ν experiments are ongoing and planned. Among them, EXO-200 [9] experiment puts an upper limit on the relevant neutrino mass matrix element |m 11 | within a range as |m 11 | < (0.14-0.38 eV). Further, NEXT-100 [10] experiment will be able to bring down the above value of the order of 0.1 eV. Thus in an optimistic point of view such property of neutrino could be testified by the next generation experiments. However, even if it is possible to pin down the value of |m 11 |, it is still difficult to predict the values of the Majorana phases until we can fix the absolute neutrino mass scale. It is shown in Ref. [11] that in addition to the ββ 0ν decay experiments, lepton number violating processes in which the Majorana phases show up are also corroborative to determine the individual Majorana phases. Another interesting physical aspect such as contribution of the Majorana phases to the generation of θ 13 within the present 3σ range of neutrino oscillation global fit data is also studied in the literature [12]. Thus it is worthwhile to study the calculability of the Majorana phases in terms of a general neutrino mass matrix (m ν ) parameters.
In the present work we estimate individual Majorana phases in terms of the parameters of a general m ν using three rephasing invariants I 12 ,I 13 and I 23 on the basis of Mohapatra-Rodejohann's phase convention [13]. We further numerically estimate the ranges of each Majorana phase for both types of hierarchies taking into account the neutrino oscillation global fit data and bound on the sum of the three light neutrino masses. This methodology is applied in the context of a cyclic symmetric model as well as a model with scaling ansatz property. The plan of the paper is as follows.
In Section 2 we briefly discuss the basic formalism to set the convention of the Majorana phase representation within the framework of neutrino oscillation phenomena. CP violating rephasing invariants are presented in Section 3. Section 4 contains explicit calculation of the Majorana phases for both types of neutrino mass hierarchies along with phenomenologically viable different sub cases. Numerical estimation of the Majorana phases in a model independent way taking into account the constraints from the extant data for both types of neutrino mass hierarchies is presented in Section 5. In Section 6 application of the above methodology in the context of a cyclic symmetric and scaling ansatz invariant models is presented. Section 7 contains summary of the present work.

Basic formalism
Experimental observation of neutrino flavour oscillation constitutes a robust evidence in favour of nonzero neutrino masses. The flavour transition process is basically a quantum mechanical interference phenomena with the explicit relationship between the left handed quantum fields (ν αL ) of the flavour basis and the mass basis (ν iL ) as where, α(= 1, 2, ...., m) corresponds to the flavour index and i(= 1, 2, ...., n) implies the mass index and the matrix U ν is the corresponding neutrino mixing matrix. For three generation of fermions, i.e, for n = m = 3, the weak Lagrangian containing charged lepton fields and the neutrino fields can be written in the mass basis as where U l is the unitary mixing matrix in the charged lepton sector.
The matrix U † l U ν is the leptonic mixing matrix and is known as the P ontecorvo − M aki − N akagawa − Sakata mixing matrix (U P M N S ) which contains 3 mixing angles and 6 phases in general. It is useful to redefine the mixing matrix by absorbing the unphysical phases into the charged lepton fields and the neutrino fields (Dirac type). If the neutrinos are Majorana type, they break the global U (1) symmetry and hence, redefinition of the neutrino fields are not possible. Therefore, out of 6 phases 3 unphysical phases can be absorbed by redefining only the charged lepton fields and thus the U P M N S matrix is parametrized as where U CKM is the usual CKM type matrix and is given by where c ij ⇒ cos θ ij ,s ij ⇒ sin θ ij and δ is the Dirac CP phase. P M is a 3 × 3 diagonal phase matrix and following Mohapatra-Rodejohann's convention [13] it is given by where α and β + δ are the Majorana phases which do not appear in the neutrino → neutrino oscillation experiments [14]. Regarding the structure of P M matrix we would like to mention the following: The advantage of using the above Majorana phase convention is that for m 3 = 0 it is possible to calculate the single existing Majorana phase α while, for m 1 = 0, only the phase difference α −(β +δ) is calculable. The result will be reversed if we utilize the PDG [16] convention. Explicitly, with PDG convention, if m 3 = 0, only the phase difference is calculable, however if m 1 is vanishing it is possible to calculate the existing Majorana phase. Based on PDG convention a detailed calculation for both the Majorana phases is presented in Ref. [17], however, if m 3 = 0 which is still allowed by the present neutrino experimental data, it is not possible to calculate individual phases in that case.
CP violating effect of Majorana phases in neutrino → antineutrino oscillation [18]- [19] and some lepton number violating (LNV) processes are studied in detail in Ref. [11]. In this work, using the rephasing invariants constructed out of the neutrino mass matrix elements [20] we determine the Majorana phases for two different hierarchical cases.

CP violating phase invariants
Considering neutrinos as the Majorana fermions in extended standard model one can parametrize U P M N S with the CP violating phases as given in Eqn. (2.3) where we redefine the charged lepton fields absorbing the unphysical phases of total mixing matrix U. Hence, in principle the mixing matrix U can be defined as where P φ is a 3 × 3 diagonal phase (unphysical) matrix and is given by Now, as the low energy neutrino mass matrix is complex symmetric it can be diagonalized as Thus P φ rotates the mass matrix m ν in phase space. Therefore, the rephasing invariants (remain invariant under phase rotation) of m ν contain the informations about the CP violating phases. It has been shown explicitly in Ref. [20] that for three generations of neutrinos there are three independent rephasing invariants and are given by where m αβ is the element of m ν at αβ position with α, β = 1, 2, 3. Now since the invariants of Eqn.(3.7) are independent of phase rotation of m ν , therefore to evaluate them in terms of mixing angles, CP violating phases and the eigenvalues we can rewrite Eqn.(3.6) as where without any loss of generality we assume φ i = 0 which corresponds to the structure of P φ as P φ = diag(1, 1, 1). Now writing down Eqn. It is now straightforward to calculate I 12 and I 13 using Eqn.(3.9) to Eqn.(3.14) and neglecting terms O(s 2 13 ) and higher order we obtain I 12 and I 13 as and with (3.23)

The Majorana phases
At the outset, first, we would like to mention that the three independent invariants I 12 , I 13 and I 23 stand for the three CP violating phases α, β + δ and δ, however in this section we solve the invariants only for the Majorana phases (α, β + δ) while the Dirac CP phase δ is calculable from the usual Jarlskog measure of CP violation. Next, if all the eigenvalues and mixing angles are nonzero, then all the invariants are independent and in principle one can extract the α and β + δ phases without any specific hierarchical assumption. However, the calculation is too cumbersome in this general situation. In the present work we consider a simplified approach assuming hierarchical structure of neutrino masses and calculate the Majorana phases utilizing the invariants I 12 ,I 13 and I 23 for both, normal and inverted hierarchical cases.  and I 23 = 0 (4.14) It is amply clear that the first two invariants I 12 and I 13 are not independent [13] of each other and their correlated relationship leads to the estimation of only one Majorana phase α while the information about the Dirac CP phase is lost.  In such a condition the three invariants are coming out in a correlated manner as

Numerical estimation
A general solution for a three generation complex symmetric Majorana mass matrix is given in Ref. [17]. In order to estimate the Majorana phases obtained in the present work we utilize the expressions of the three eigenvalues and the three mixing angles. We also use the global fit data of neutrino oscillation experiments shown in Table 1 and the upper limit on the sum of the neutrino masses (Σ i m i < 0.23eV ) to obtain model independent ranges of the Majorana phases. In our We consider a most general 3 × 3 complex symmetric neutrino mass matrix m ν as where a i ,b i ' s are Lagrangian parameters. The three rephasing invariants are coming out in terms of the elements of m ν as Upon numerical estimation the model independent ranges for α and β + δ come out as −78 o < α < 77.

Examples with symmetry approach
The above numerical results are obtained for the general m ν where all the 12 independent parameters are present. However, one can reduce the number of parameters by invoking some symmetry or ansatz in the Lagrangian. Latter, we provide applications of the general result in few typical cases for both the hierarchies, normal and inverted.

Normal hierarchy
In this case we explore a model that corresponds to Case I of the normal hierarchical scenario mentioned in section (4.2). The model is based on cyclic symmetry with type I seesaw mechanism to accommodate the neutrino oscillation data. In the fundamental level the symmetry exists in the neutrino sector of the Lagrangian and due to the symmetry a degeneracy in masses occurs removal of which therefore requires breaking of the symmetry. It is shown that a minimal breaking in the Majorana mass matrix is sufficient to explain the extant data. In this model broken symmetric mass matrix m ν (= −m D M −1 R m T D ) due to type I seesaw mechanism is given by In addition, we also plot the variation of the rephasing invariants with the lightest neutrino mass (m 1 ) and one can see the ascending nature of the value of the invariants with the increment of the lightest neutrino mass.

Inverted hierarchy
In this case , we explore a model based on scaling ansatz with inverse seesaw mechanism [21]- [30]. In this mechanism m ν is given by where m D is the usual Dirac type matrix and the other two matrices µ (Majorana type) and M RS (Dirac type) arise due to the interaction between the additional singlet fermion and right handed neutrino considered in this type of seesaw mechanism. To further reduce the number of parameters texture zeros [31]- [59] are assumed in the constituent m D and µ matrices. Scaling ansatz invariance dictates m 3 = 0 and θ 13 = 0 and this case corresponds to Case III of sec.(4.1). Thus to generate non zero θ 13 breaking of the ansatz is necessary. Incorporating breaking in m D through a small parameter , there are two different phenomenologically survived textures which are given by and where all the parameters are complex and follow their usual definitions given in Ref. [60]. In both the cases θ 13 = 0 however, m 3 = 0 due to singular nature of µ matrix and this case corresponds to Case II of sec.(4.1). We further consider the most general version of the above case through the breaking of the ansatz in both m D and µ matrices through two small parameters and respectively and the neutrino mass matrix m 3 ν comes out as and in this situation both θ 13 and m 3 are nonzero corresponding to Case I of sec.(4.1). Thus the whole inverted hierarchical sector is generated through the choice of the above model. Now, with the explicit construction of rephasing invariants we calculate the Majorana phases in each case. Interestingly, for all the cases, the value of J CP comes out very small due to smallness of the Dirac CP phase δ, or more precisely almost real nature of the mass matrices. Therefore, such typical nature of the mass matrices also constrain the Majorana phases approximately as −1. Finally, to summaries the numerical results we present table that shows the ranges of the Majorana phases in each cases.

Summary
In the present work we calculate the Majorana phases of a general complex symmetric 3 × 3 neutrino mass matrix utilizing the three rephasing invariant quantities I 12 ,I 13 and I 23 for both the hierarchical structures of neutrinos using Mohapatra-Rodejohann's phase convention. Such methodology enables us to estimate the existing Majorana phase even if one of the eigenvalue (m 3 ) is zero. However, if m 1 = 0, this methodology will only enable us to calculate the difference of the Majorana phases. In that case to evaluate the remaining Majorana phases we have to utilize the methodology presented in Ref. [17] based on PDG phase convention. We further exemplify the above procedure in few typical cases leading to normal and inverted hierarchy. For normal hierarchical case we give an example of a model based on cyclic symmetry within type I seesaw mechanism. We estimate the Majorana phases for the broken symmetric case, since cyclic symmetry dictates a degeneracy in the mass eigenvalues. As an example of inverted hierarchy, we cite a model comprises of scaling ansatz, texture zeros and inverse seesaw mechanism. Different sub cases of inverted hierarchy as mentioned in the present work are obtained depending upon the scheme of incorporation of ansatz breaking mechanism. A typical characteristic of this case is that the value of J CP as well as δ is vanishingly small due to almost real nature of neutrino mass matrix and thereby leading to a narrow range of the Majorana phases.