The Texture One Zero Neutrino Mass Matrix With Vanishing Trace

In the light of latest neutrino oscillation data, we have investigated the one zero Majorana neutrino mass matrix $M_{\nu}$ with zero sum condition of mass eigen values in the flavor basis, where charged lepton mass matrix is diagonal. Among the six possible one-zero cases, it is found that only five can survive the current experimental data, while case with (1, 1) vanishing element of $M_{\nu}$ is ruled out, if zero trace condition is imposed at 3$\sigma$ confidence level (CL). Numerical and some approximate analytical results are presented.


Introduction
The Double Chooz, Daya Bay and RENO Collaborations [1][2][3], have finally established the non-zero and relatively large value of the reactor mixing angle θ 13 , hence the number of known available neutrino oscillation parameters approaches to five viz. two mass squared differences (δm 2 , ∆m 2 ) and three neutrino mixing angles (θ 12 , θ 23 , θ 13 ).
However, any general 3 × 3 neutrino mass matrix contains more parameters than can be measured in realistic experiments. In fact, assuming the Majorana type nature of neutrinos, the neutrino mass matrix contains nine real free parameters: three neutrino masses (m 1 , m 2 , m 3 ), three flavor mixing angles (θ 12 , θ 23 , θ 13 ) and three CP violating phases (δ, ρ, σ).
In order to reduce the number of free parameters, several phenomenological ideas, in particular texture zeros [4][5][6][7][8][9][10][11][12] have been widely adopted in the literature. The imposition of texture zeros in neutrino mass matrix leads to some important phenomenological relations between flavor mixing angles and fermion mass ratios [10][11][12]. In the flavor basis, where charged lepton mass matrix is diagonal, at the most two zeros are allowed in neutrino mass matrix, which are consistent with neutrino oscillation data [12]. The analysis of two texture zero neutrino mass matrices limits the number of experimentally viable cases to seven. The phenomenological implications of one texture zero neutrino mass matrices have also been studied in the literature [7][8][9] and it has been observed that all the six cases are viable with experimental data. However, the imposition of single texture zero condition in neutrino mass matrix makes available larger parametric space for viability with the data compared with texture two zero. In order to impart predictability to texture one zero, additional constraints in the form of vanishing determinant [13] or trace can be incorporated. The phenomenological implication of determinantless condition on texture one zero have been rigoursly studied in Refs. [7,13,14]. The implication of traceless condition, was first put forward in [15] wherein the anomalies of solar and atmospheric neutrino oscillation experiments as well as the LSND experiment were simulataneously explained in the framework of three neutrinos. In Ref. [16], X. G. He and A. Zee have particulary investigated the case of CP conserving traceless neutrino mass matrix for explaining the solar and atmospheric neutrino deficits. Further motivation of traceless mass matrices can be provided by models wherein neutrino mass matrix can be constructed through a commutator of two matrices, as it happens in models of radiative mass generation [17]. In Ref. [18], H. Alhendi et. al. have studied the case of two tracless submatrices of Majorana mass matrix in the flavor basis and carried out a detailed numerical analysis at 3σ confidence level. The phenomenological implications of traceless neutrino mass matrix on neutrino masses, CP violating phases and effective neutrino mass term is also studied in Ref. [19], for both normal and inverted mass ordering and in case of CP conservation and violation, respectively. In the present work we impose the traceless condition on texture one zero Majorana mass matrix and investigate the outcomes of such condition on the parameteric space of neutrino masses (m 1 , m 2 , m 3 ) and CP violating phases (δ, ρ, σ).
Assuming the Majorana nature of neutrinos, neutrino mass matrix is complex symmetric. In the flavor basis, if one of the element is considered to be zero, the number of possible cases turns out to be six, which are given below T 1 : (1) where '× ' stands for non-zero element and complex matrix element.
Among these possible cases, there exists a permutation symmetry between certain pair of cases viz. (T 2 , T 3 ) and (T 4 , T 5 ), while cases T 1 and T 6 tranform onto themselves independently. The origin of permutation symmetry is explained from the fact that these pairs are related by exchange of 2-3 rows and 2-3 columns of neutrino mass matrix. The corresponding permutation matrix is given by which leads to the following relations among the neutrino oscillation parameters where X and Y superscripts denote the cases related by 2-3 permutation symmetry.
The rest of the work is planned as follows: In Section 2, we discuss the methodology used to reconstruct the Majorana neutrino mass matrix and subsequently obtain some useful phenomenological relations of neutrino mass ratios and Majorana phases by incorporating texture one zero and zero trace conditions simultaneously.
In Section 3, we present the numerical analysis using some approximate analytical relations. In Section 4, we summarize our work.

Formalism
In the flavor basis, the Majorana neutrino mass matrix M ν , depending on three neutrino masses (m 1 , m 2 , m 3 ) and the flavor mixing matrix can be expressed as, The mixing matrix V can be written as V = U P , where U denotes the neutrino mixing matrix consisting of three flavor mixing angles and one Dirac-like CP violating phase, whereas, the matrix P is a diagonal phase matrix, i.e., P =diag(e iρ , e iσ , 1) with ρ and σ being the two Majorana CP violating phases. The neutrino mass matrix M ν can then be rewritten as where λ 1 = m 1 e 2iρ , λ 2 = m 2 e 2iσ , λ 3 = m 3 .
For the purpose of calculations, we have adopted the parameterization of the mixing matrix U considered by Ref. [5], e.g., where c ij = cosθ ij , s ij = sinθ ij for i,j=1,2,3 and δ is the CP violating phase.
If one of the elements of M ν is considered zero, i.e. M lm = 0, it leads to the following constraint equation where l, m run over e, µ and τ .
The traceless condition implies sum of the mass eigen values in neutrino mass matrix is zero i.e.
Using Eqs. 7 and 8, we obtain The magnitudes of neutrino mass ratios are given by  Table 1: Current neutrino oscillation parameters from global fits at 1σ, 2σ and 3σ confidence level [21]. NO(IO) refers to normal (inverted) neutrino mass ordering.
The expressions for three neutrino masses (m 1 , m 2 , m 3 ) can be given as Thus the neutrino mass spectrum can be fully determined.
The expression for Jarlskog rephasing parameter J CP , which is a measure of CP violation, is given by

Numerical results and Discussion
The experimental constraints on neutrino parameters at 1σ, 2σ and 3σ confidence level (CL) are given in Table   1.
The effective Majorana mass term relevant for neutrinoless double beta (0νββ) decay is given by The future observation of 0νββ decay would imply lepton number violation and Majorana character of neutrinos.
For recent reviews see Refs. [22,23]. There are large number of projects such as CUORICINO [24], CUORE [25], GERDA [26], MAJORANA [27], SuperNEMO [28], EXO [29], GENIUS [30] which target to achieve a sensitivity upto 0.01eV for |M | ee . For the present analysis, we assume the upper limit on |M | ee to be less than 0.5eV at 3σ CL [23]. The data collected from the Planck satellite [31] combined with other cosmological data put a limit on the sum of neutrino masses as Here, we take rather more conservative limit on sum of neutrino masses (Σ) (i.e. Σ < 1eV) at 3σ CL. We span the parameter space of input neutrino oscillation parameters (θ 12 , θ 23 , θ 13 , δm 2 , ∆m 2 ) by choosing the randomly generated points of the order of 10 6−7 . Using Eq. 15, the parameter space of CP-violating phases (δ, ρ, σ), effective mass term |M | ee , neutrino masses (m 1 , m 2 , m 3 ) can be subsequently constrained. In order to interpret the phenomenological results, some approximate analytical relations (up to certain leading order term of s 13 ) have been used in the following discussion. The exact analytical relations of neutrino mass ratios (ξ, ζ) have been provided in Table 2.

Case T 1
Using Eqs. 6, 9 and 10, in the leading order term of θ 13 , we obtain the following analytical relations For NO, using Eq. 15, we obtain R ν ≈ ζ 2 − ξ 2 ≈ sec2θ 12 . Using 3σ experimental range of oscillation parameters, we find, 2.23 ≤ R ν ≤ 4.02, which excludes the experimental range of R ν and for IO , we have which is again inconsistent with current experimental data as R ν > 0.75. Therefore, case T 1 is ruled out with the latest neutrino oscillation data at 3σ CL.
It is found from the analysis that case T 2 favors both normal (NO) and inverted mass ordering (IO) at 3σ CL.
The parameter space of CP violating phases δ, ρ, σ is found to be constrained to very small ranges for NO [ Fig.   1(a, b)]. However, for IO, comparatively significant allowed parameter space is available for δ, ρ, σ [ Fig. 2(a, b)].
In the leading order approximation of s 13 , the effective mass term in 0νββ decay turns out to be which lies well within the sensitivity limit of neutrinoless double beta decay experiments. Fig. 1(c) and Fig. 2(c) show the correlation plot between |M | ee and δ for NO and IO respectively. The Jarlskog rephrasing parameter J CP is found to be non vanishing for NO [ Fig. 1(d)], however, J CP = 0 cannot be excluded for IO [ Fig. 2(d)].

Case T 3
With the help of Eqs. 6, 9 and 10, we deduce the following analytical expressions in the leading order of s 13 term.
and the Majorana CP violating phases Cases T 2 and T 3 are related via permutation symmetry, therefore the phenomenological results for case T 3 can be obtained from case T 3 by using Eq. 3. The correlation plots for case T 3 have been complied in Fig. 3(a, b, c, d) (NO) and Fig. 4(a, b, c, d) (IO), indicating the parameter space of ρ, σ, δ, |M | ee , J CP .

Case T 4
Using Eqs. 6, 9 and 10, we deduce the following analytical expressions in the leading order of s 13 term Since R ν = 0 in the leading order approximation of s 13 , we have to work at next to leading order, and we get Using the best fit values from latest global fits on neutrino oscillation data [ Table 1], the neutrino mass spectrum can be given as follows implying that only NO is allowed. Fig. 5(a, b) show the correlation plot between Majorana phases (ρ, σ) and Dirac CP violating phase (δ). The parameter space for δ is found to be restricted near 90 0 and 270 0 . The prediction is significant considering the latest hint on δ near 270 0 in the recent global fits on neutrino oscillation data [ Table 1]. The Majorana phases (ρ, σ) are found to be constrained near −90 0 and 90 0 . In Fig. 5(d), it is explicitly shown that J CP is non-zero implying that Case T 5 is necessarily CP violating.
In the leading order of s 13 , the effective mass term in 0νββ decay can be approximated as which is well within the accessible limit of next generation neutrinoless double decay experiments. The correlation plot between |M | ee and δ has been provided for case T 4 in Fig. 5(c).

Case T 5
With the help of Eqs. 6, 9 and 10, we obtain the following analytical expressions in the leading order of s 13 term Since R ν = 0 in the leading order approximation of s 13 , we have to work at next to leading order, and we obtain Using the best fits from latest global neutrino oscillation data, the neutrino mass spectrum can be given as follows indicating that only NO is allowed. Since T 4 and T 5 are related due to permutation symmetry, therefore their phenomenological implications are similar. The phenomenological results for case T 5 can be derived from case T 4 using Eq. 3. The correlation plots for ρ, σ, δ, |M | ee , J CP have been complied in Fig. 6.
In the leading order of s 13 term, the effective mass term in 0νββ decay can be approximated as
From the analysis, out of six cases, only T 1 is found to be inconsistent with the experimental data for both normal and inverted mass ordering. For remaining cases the parameter space of CP violating phases (δ, σ, ρ), effective mass term |M | ee , neutrino masses (m 1 , m 2 , m 3 ) is found to be constrained to an appreciable extent at 3σ CL. The allowed ranges of all the five viable cases for Dirac CP violating phase (δ), Majorana phases (ρ, σ) and neutrino masses (m 1 , m 2 , m 3 ) have been summarized in Table 3.

Summary and conclusion
In the present work, we have systematically analyzed the texture one zero Majorana mass matrix along with zero trace condition. In our analysis, we find that case T 1 with vanishing (1, 1) element of M ν is ruled out with current experimental data. Therefore, out of six possible cases of texture one zero with zero trace, only five viz.