Advances in High Energy Physics

Volume 2018, Article ID 2863184, 13 pages

https://doi.org/10.1155/2018/2863184

## The Texture One-Zero Neutrino Mass Matrix with Vanishing Trace

Department of Physics, National Institute of Technology Kurukshetra, Haryana 136119, India

Correspondence should be addressed to Madan Singh; moc.liamg@971nadamhgnis

Received 14 October 2017; Accepted 6 December 2017; Published 5 April 2018

Academic Editor: Andrzej Okniński

Copyright © 2018 Madan Singh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

In the light of latest neutrino oscillation data, we have investigated the one-zero Majorana neutrino mass matrix with zero sum condition of mass eigenvalues 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 is ruled out, if zero trace condition is imposed at confidence level (CL). Numerical and some approximate analytical results are presented.

#### 1. Introduction

The Double Chooz, Daya Bay, and RENO Collaborations [1–3] have finally established the nonzero and relatively large value of the reactor mixing angle ; hence the number of known available neutrino oscillation parameters approaches five, namely, two mass-squared differences and three neutrino mixing angles . However, any general 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 , three flavor mixing angles , and three CP violating phases .

In order to reduce the number of free parameters, several phenomenological ideas, in particular texture zeros [4–26], 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 [24–27]. In the flavor basis, where charged lepton mass matrix is diagonal, at most two zeros are allowed in neutrino mass matrix, which are consistent with neutrino oscillation data [25, 26]. 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 matrix have also been studied in the literature [20–23] 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 larger parametric space for viability with the data available compared with two-zero texture. In order to impart predictability to one-zero texture, additional constraints in the form of vanishing determinant [28] or trace can be incorporated. The phenomenological implication of determinantless condition on one-zero texture have been rigorously studied in [20, 28–32]. The implication of traceless condition was first put forward in [33] wherein the anomalies of solar and atmospheric neutrino oscillation experiments as well as the LSND experiment were simultaneously explained in the framework of three neutrinos. In [34], Zee has particularly 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 what happens in models of radiative mass generation [35]. In [36], Alhendi et al. have studied the case of two trackless submatrices of Majorana mass matrix in the flavor basis and carried out a detailed numerical analysis at confidence level. The phenomenological implications of traceless neutrino mass matrix on neutrino masses, CP violating phases, and effective neutrino mass term are also studied in [37], 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 parametric space of neutrino masses and CP violating phases .

Assuming the Majorana nature of neutrinos, neutrino mass matrix is complex symmetric. In the flavor basis, if one of the elements is considered to be zero, the number of possible cases turns out to be six, which are given below:where “” stands for nonzero element and complex matrix element.

Among these possible cases, there exists a permutation symmetry between certain pair of cases, namely, and , while cases and transform 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 bywhich leads to the following relations among the neutrino oscillation parameters:where and 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.

#### 2. Formalism

In the flavor basis, the Majorana neutrino mass matrix , depending on three neutrino masses and the flavor mixing matrix can be expressed asThe mixing matrix can be written as , where denotes the neutrino mixing matrix consisting of three flavor mixing angles and one Dirac-like CP violating phase, whereas the matrix is a diagonal phase matrix; that is, = diag with and being the two Majorana CP violating phases. The neutrino mass matrix can then be rewritten aswhere , , and

For the purpose of calculations, we have adopted the parameterization of the mixing matrix considered by [5]; for example,where , for , and is the CP violating phase.

If one of the elements of is considered zero, that is, , it leads to the following constraint equation:where , run over , , and .

The traceless condition implies that sum of the mass eigenvalues in neutrino mass matrix is zero; that is,Using (7) and (8), we obtainThe magnitudes of neutrino mass ratios are given byUsing (9), we find the following analytical relations for Majorana phases :Thus neutrino mass ratios and two Majorana-type CP violating phases can fully be determined in terms of three mixing angles and the Dirac-type CP violating phase . The ratio of two neutrino mass-squared differences in terms of neutrino mass ratios and is given bywhere and [38] corresponds to solar and atmospheric neutrino squared differences, respectively. The sign of is still not known experimentally; that is, or corresponds to the normal or inverted mass ordering of neutrinos.

The expressions for three neutrino masses can be given asThus the neutrino mass spectrum can be fully determined.

The expression for Jarlskog rephasing parameter , which is a measure of CP violation, is given by

#### 3. Numerical Results and Discussion

The experimental constraints on neutrino parameters at , , and confidence level (CL) are given in Table 1.