Phenomenology of Trimaximal Mixing with One Texture Equality

Dev, S.; Raj, Desh

doi:https://doi.org/10.1155/2022/4952562

Advances in High Energy Physics

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Abstract Introduction Data Availability Disclosure Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 4952562 | https://doi.org/10.1155/2022/4952562

Phenomenology of Trimaximal Mixing with One Texture Equality

S. Dev¹and Desh Raj²

Academic Editor: Omar G. Miranda

Received21 Mar 2022

Revised27 Jul 2022

Accepted06 Sept 2022

Published30 Sept 2022

Abstract

We study neutrino mass matrices with one texture equality and the neutrino mixing matrix having either its first () or second () column identical to that of the tribimaximal mixing matrix. We found that out of total fifteen possible neutrino mass matrices with one texture equality, only six textures are compatible with mixing and six textures are compatible with mixing in the light of the current neutrino oscillation data. These textures have interesting implications for the presently unknown parameters such as the neutrino mass scale, effective Majorana neutrino mass, effective neutrino mass, the atmospheric mixing, and the Dirac- and Majorana-type CP violating phases. We, also, present the group motivation for some of these textures.

1. Introduction

In the last two decades, significant advances have been made by various neutrino oscillation experiments in determining the neutrino masses and mixings. Various neutrino parameters like three mixing (solar, atmospheric, and reactor) angles and the two mass squared differences ( and ) have been measured by various neutrino oscillation experiments with fairly good precision. In addition, the recent neutrino oscillation data hint towards a nonmaximal atmospheric mixing angle () [1] and Dirac-type CP-violating phase () near 270° [2, 3]. However, many other attributes like leptonic CP-violation, neutrino mass ordering (normal mass ordering (NO) or inverted mass ordering (IO)), nature of neutrinos (Dirac or Majorana), and absolute neutrino mass scale are still unknown. Furthermore, the origin of the lepton flavor structure still remains an open issue. The neutrino mass matrix which encodes the neutrino properties contains several unknown physical parameters. The phenomenological approaches based on Abelian or non-Abelian flavor symmetries can play a significant role in determining the specific texture structure of the neutrino mass matrix with reduced number of independent parameters.

Several predictive models such as texture zeros [4–25], vanishing cofactors [26–37], equalities among elements/cofactors [38, 39], and hybrid textures [40–46] amongst others can explain the presently available neutrino oscillation data, since the presence of texture equalities, just like texture zeros or vanishing cofactors, reduces the number of free parameters in the neutrino mass matrix and, hence, must have a similar predictability as that of texture zeros or vanishing cofactors. In the flavor basis, neutrino mass matrices with one texture equality and two texture equalities have been studied in the literature [38, 39]. The hybrid textures which combine a texture equality with a texture zero or a vanishing cofactor have been studied in the literature [40–46].

In addition, discrete non-Abelian symmetries leading to the Tri-Bi-Maximal (TBM) [47, 48] neutrino mixing pattern have been widely studied in the literature. The TBM mixing matrix given by predicts a vanishing reactor mixing angle () and maximal atmospheric mixing angle (), and the solar mixing angle is predicted to be which is. However, the nonzero value of confirmed by various neutrino oscillation experiments underlines the need for necessary modifications to the TBM mixing pattern to make it compatible with the present experimental data [49–53]. One of the simplest possibilities is to keep one of the columns of the TBM mixing matrix unchanged while modifying its remaining two columns to within the unitarity constraints. This gives rise to three mixing patterns, viz., , , and which have their first, second, and third columns identical to the TBM mixing matrix, respectively. The mixing scheme predicts and is, hence, phenomenologically unviable. The and mixing schemes have been successfully employed to explain the pattern of lepton mixing and have been extensively studied in the literature [54–71]. The mixing, in particular, gives a very good fit to the present neutrino oscillation data. Recently, neutrino mass matrices with texture zero(s) in combination with and mixing have been studied [72–74].

In the present work, we study a class of neutrino mass matrices having one texture equality with or of the TBM in the neutrino mixing matrix. Neutrino mass matrices having one texture equality along with or of the TBM have a total of six free parameters and, hence, lead to very predictive textures for the neutrino mass matrices.

There are a total of fifteen possible structures with one texture equality in the neutrino mass matrix, and they are listed in Table 1.

There exists a - permutation symmetry between different structures of neutrino mass matrices, and the corresponding permutation matrix has the following form:

Neutrino mass matrices with one texture equality, therefore, are related to each other as leading to the following relations between the neutrino oscillation parameters:

Neutrino mass matrices with one texture equality related by the - permutation operation are

In the flavor basis, where the charged lepton mass matrix is diagonal, the complex symmetric Majorana neutrino mass matrix can be diagonalized by a unitary matrix: where. The unitary matrix can be parametrized as

where with, . is the diagonal phase matrix containing the two Majorana-type CP-violating phases and . is the Dirac-type CP-violating phase. The phase matrix is physically unobservable. The matrix is called the neutrino mixing matrix or the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) [75–78] matrix. The effective Majorana neutrino mass matrix can be written as

The Dirac-type CP-violation in neutrino oscillation experiments can be described in terms of the Jarlskog rephasing invariant quantity [79] with

The effective Majorana neutrino mass , which determines the rate of neutrinoless double beta decay, is given by

There are many experiments such as CUORICINO [80], CUORE [81], MAJORANA [82], SuperNEMO [83], and EXO [84] which aim to achieve a sensitivity up to 0.01 eV for . The KamLAND-Zen experiment [85] provides the upper limits on the effective Majorana neutrino mass which is given by at 90% confidence level (C.L.).

The measurement of the absolute neutrino mass scale via the decay kinematics is usually described by the effective neutrino mass [86]

Recently, the KATRIN [87] experiment has reported the upper limit of at 90% C.L.

Further, cosmological observations provide more stringent constraints on absolute neutrino mass scale by putting an upper bound on the sum of neutrino masses:

Recent Planck data [88] in combination with baryon acoustic oscillation (BAO) measurements provide a tight bound on the sum of neutrino masses eV at 95% C.L.

2. Mixing and One Texture Equality

A neutrino mass matrix with mixing can be written as where the mixing matrix , also known as trimaximal mixing, can be parametrized [68–71] as

The mass matrix is invariant under the transformation with , as the generator of symmetry [90, 91]. Invariance of under when combined with one texture equality leads to the equality of three unphysical phases in , i.e., .

The most general neutrino mass matrix with as the mixing matrix can be parametrized as

The neutrino mass matrix can be realized within the framework of an model where the flavor symmetry is spontaneously broken by two real triplets, and three real singlets, which are gauge singlets [92–94]. Upon symmetry breaking, the VEVs of the flavone singlets and triplets take the alignments

The neutrino mass matrix, in the flavor basis, is given by

The above mass matrix leads to the neutrino mixing matrix. For , the above mass matrix leads to the TBM neutrino mixing matrix. The equality among the elements of mass matrix in Eq. (17) does not arise naturally, and hence, we assume additional constraints on the elements of mass matrix, e.g., which leads to one equality between the and elements of in Eq. (17). Therefore, all possible textures of neutrino mass matrices with mixing and one texture equality are given by

where the neutrino mass matrices in each equation are related by - symmetry. The neutrino mixing angles can be calculated by using the following relations:

Substituting the elements of form Eq. (16) into Eq. (26), we get

Using Eqs. (10) and (16), the Jarlskog rephasing invariant is given by and the Dirac-type CP-violating phase can be calculated by using the equation [72–74]

From Eqs. (11) and (16), the effective Majorana mass for mixing is given by and the effective neutrino mass for mixing can be calculated by using Eqs. (12) and (16) as

The existence of one equality between the elements and of the neutrino mass matrix implies which yields the complex equation where and is the PMNS matrix given in Eq. (7). The above equation can be rewritten as where with and can take values , and . Since the mixing has equal elements in the second column, it leads to . Therefore, using Eq. (16) in Eq. (34), we have

Simultaneous solution of the real and imaginary parts of Eq. (36) leads to

Using experimentally available mass squared differences and for NO (IO) with Eq. (37), the three neutrino mass eigenvalues are given by

where , for NO, and for IO.

For the numerical analysis, we generate - points. The mass squared differences and for NO (IO) are varied randomly within their 3 experimental ranges given in Table 2. Parameters , and are, also, varied randomly within their full ranges (0-90°), (0-360°), and (0-360°), respectively. Equation (38), (39), and (40) are used to calculate the Majorana-type CP-violating phase and three mass eigenvalues (, and ) for both mass orderings. In addition, the mixing angles , and are calculated by using Eq. (27) and must satisfy the experimental data given in Table 2. The Jarlskog invariant (), Dirac-type CP-violating phase (), effective Majorana mass (), effective neutrino mass (), and the sum of neutrino masses () are calculated by using Eqs. (28), (29), (30), (31), and (14), respectively.

The numerical predictions for various neutrino parameters are given in Tables 3 and 4. Table 3 provides numerical predictions for viable textures under the constrains from neutrino oscillation data, whereas Table 4 provides numerical predictions for viable textures under the constrains from cosmological and neutrinoless double beta decay bounds along with neutrino oscillation data. The allowed range of parameter is (0-360°) for all viable textures. which lies in the range (−0.037-0.037) for all viable textures except with NO (IO), and for these textures, the range of is ±(0.011–0.037). The parameter is constrained to lie within the ranges (10.0°-11.1°) for all viable textures. The solar mixing angle () is constrained to lie in the ranges (35.68°-35.77°) for all allowed textures. Figures 1 and 2 show correlations among various neutrino oscillation parameters. The Dirac CP-violating phase and phase are linearly correlated as shown in Figure 2(e). Figure 1(h) depicts the correlation between and . The Dirac-type CP-violating phase strongly depends on the Majorana-type CP-violating phase as shown in Figure 1(a).

The main results for the neutrino mass matrices with one texture equality and mixing are listed in the following: (i)Textures , and lead to two degenerate eigenvalues and are, hence, experimentally ruled out at C.L.(ii)Textures , and lead to vanishing reactor mixing angle and, hence, are not viable at C.L.(iii)Textures and for IO are not consistent with the experimental data if cosmological and neutrinoless double beta decay bounds along with neutrino oscillation data are incorporated(iv)Textures and are consistent with NO only, whereas textures and are consistent with IO only(v)For NO, textures and are not consistent with the experimental data as the mixing angles and are not within the range(vi)All viable textures cannot have zero lowest mass eigenvalue for both mass orderings(vii)The atmospheric mixing angle is below (above) maximal for textures () and () with NO and IO, respectively(viii) is maximal for or for textures () and () with NO (IO).(ix)The parameter is found to be nonzero for all viable textures except and. gets its largest value when or for textures and .(x)For all viable textures, the effective neutrino mass () is well within the range provided by KATRIN experiment(xi)The parameters, , and get their largest value when for textures and with NO (IO).

3. Group Motivation

The , permutation group of three objects, is the smallest discrete non-Abelian group. The permutation matrices in the three dimensional reducible representation are

where matrices in each equation belong to the same class of . The most general neutrino mass matrix invariant under the group is proportional to the democratic matrix and is given by where is a complex number and is called the Democratic matrix. The exact symmetry does not satisfy the current neutrino oscillation data, and hence, symmetry must be broken. Various models based on the symmetry have been presented in Refs. [95–122]. In Ref. [123, 124], the symmetry is broken by the linear combination of group matrices and successfully generates the nonzero .

The mass matrices in Eqs. (20)–(25) can be seen as the linear combination of a democratic part and a symmetry breaking part. The symmetry breaking matrix is the sum of two symmetric matrices out of which one is the group matrix which can be any of the matrices, and the other part is chosen in such a way that the resultant neutrino mass matrix still satisfies the magic symmetry [125–129] and remains invariant under symmetry. The mass matrix can be rewritten as where and are arbitrary parameters with .The symmetry of the neutrino mass matrix is broken, and the resultant neutrino mass matrix still satisfies invariant constraints

This leads to a trimaximal eigenvector for the resultant neutrino mass matrix. For example, a typical form of neutrino mass matrix is given by

In this analysis, we take the charged lepton mass matrix to be diagonal. If a horizontal symmetry exists, it must, simultaneously, be a symmetry of the neutrinos as well as the charged leptons before the gauge symmetry breaking. After the symmetry breaking when the fermions acquire nonzero masses, the neutrino sector and the charged lepton sector should be governed by different subgroups of the symmetry group in order to have nonzero mixing. Here, we consider to be the residual symmetry in the neutrino sector and symmetry as the residual symmetry in the charged lepton sector which yields nondegenerate diagonal charged lepton mass matrix [130].

Similarly, other viable textures in Eqs. (20)–(22) can, also, be decomposed into the democratic invariant part and the symmetry breaking part. The phenomenologically viable mass matrices in Eqs. (20)–(22) are related as follows by permutation symmetry:

4. Mixing and One Texture Equality

The neutrino mixing matrix with first column identical to TBM can be parametrized [60–67] as and the corresponding neutrino mass matrix is given by

The most general neutrino mass matrix with mixing can be written as

The mass matrix is invariant under the transformation where is the generator of symmetry. This along with equality condition restricts the three unphysical phase angles to .

All possible textures of neutrino mass matrices with mixing and one texture equality are given by

where textures represented in each equation are related by - permutation symmetry. The neutrino mixing angles for mixing in terms of parameters and are [72–74] given by

For mixing, the Jarlskog rephasing invariant [72–74] is and the Dirac-type CP-violating phase [72–74] is given by

The effective Majorana mass for mixing can be calculated by using Eqs. (11) and (48) as and the effective neutrino mass for by using Eqs.(12) and (48) is given by

The existence of one equality between the elements of the neutrino mass matrix implies which yields the following complex equation: where . The above equation can be rewritten as where with and can take values , and . Solving the real and imaginary parts of Eq. (59) simultaneously, we obtain the following two mass ratios:

These mass ratios can be used to calculate the ratio of mass squared differences () which is given by for NO and IO, respectively. Since, and for NO (IO) are experimentally known, the parameter should lie within its experimentally allowed range for a texture equality to be compatible with the current neutrino oscillation data. The neutrino mass eigenvalues can be calculated by using the relations for NO and IO, respectively.

For numerical analysis, we follow the same procedure as in mixing except that the parameters and are generated randomly within their allowed ranges. The mass eigenvalues are calculated by using Eq. (63), and texture equality is imposed by requiring the parameter in Eq. (62) to lie within its experimental range.

The numerical predictions for unknown parameters are summarized in Table 5 (where constrains only from neutrino oscillation data are used) and Table 6 (where the constrains from cosmological and neutrinoless double beta decay bounds along with neutrino oscillation data are used). The allowed ranges of the parameter are (-) for all viable textures. The parameter is constrained to lie in the ranges (-), whereas lies in the ranges for all viable textures. The Majorana phase varies in the range (-) for all viable textures with NO only. Correlation plots among various neutrino oscillating parameters are shown in Figures 3 and 4 for NO and IO, respectively. strongly depends on the Majorana phase as shown in Figure 3(a) for NO and Figure 4(b) for IO. As shown in Figure 3(d), is inversely proportional to which is the classical prediction of mixing. The Dirac-type CP-violating phase is constrained to lie in the regions around 90° and 270° which is consistent with the recent observations in the long-baseline neutrino oscillation experiments such as T2K and NOvA [2, 3] which shows a preference for the Dirac-type CP-violating phase to lie around . The main implications for textures having mixing with one texture equality are summarized in the following: (i)For IO, textures, , and are not consistent with the neutrino oscillation data at C.L.(ii)For NO, textures and predict large and small and are, hence, experimentally ruled out at C.L.(iii)Textures and predict a vanishing reactor mixing angle and degenerate mass eigenvalues and are, hence, not viable for both mass orderings(iv)Textures , , , , , and for both mass orderings and textures , for NO predict large and are, hence, not viable with experimental data when cosmological and neutrinoless double beta decay bounds along with neutrino oscillation data are incorporated(v)All viable textures cannot have zero lowest mass eigenvalue for both mass orderings(vi)The atmospheric mixing angle is maximal for or for all viable textures(vii)The parameter is found to be bounded from below for all viable textures except and with NO(viii)The parameter is found to lie within the current experimental range for all viable textures(ix)The Dirac-type CP-violating phase is directly proportional to the parameter for all viable textures

5. Summary

We studied the phenomenological implications of one texture equality in the neutrino mass matrix with or mixing. The presence of one texture equality in with or as the mixing matrix reduces the number of free parameters significantly and, hence, leads to very predictive neutrino mass matrices. Out of total fifteen possible textures of , thirteen textures are phenomenologically allowed with mixing, and only six textures are allowed with mixing in the light of current neutrino oscillation data at C.L. However, the number of viable textures reduced to six for mixing if the constrains from cosmology and neutrinoless double beta decay experiments along with neutrino oscillation data are used. Since, the mixing predicts a value of away from its best fit value, mixing is phenomenologically more appealing. In this analysis, we have obtained interesting predictions for unknown parameters such as the Dirac- and Majorana-type CP-violating phases, effective Majorana neutrino mass, effective neutrino mass, Jarlskog rephasing invariant, neutrino mass scale, and the sum of neutrino masses. For mixing, the Dirac-type CP-violating phase is restricted to the regions around and , the atmospheric mixing angle is maximal for or , and the lowest neutrino mass eigenvalue cannot be zero for all viable textures. For mixing, the CP-violating phases and are strongly correlated, is below (above) maximal for textures and with NO and IO, respectively, and the lowest mass eigenvalue cannot be zero for all viable textures. For mass matrices with one texture equality, the residual symmetry is broken, and the resulting neutrino mass matrix is invariant under 64.

Data Availability

The numerical prediction data used to support the findings of this study are included within the article. The raw data used to support the findings of this study are available from the corresponding author upon request.

Disclosure

A previous version of this manuscript was presented in http://arXiv.org with identifier 2202.13070.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The research work of S. D. is supported by the Council of Scientific and Industrial Research, India, via grant no. 03 (1333)/15/EMR-II. S. D. gratefully acknowledges the kind hospitality provided by IUCAA, Pune. The authors thank Radha Raman Gautam and Lal Singh for carefully reading the manuscript.

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Copyright © 2022 S. Dev and Desh Raj. 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 SCOAP³.

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