One vanishing minor in neutrino mass matrix using trimaximal mixing

Evidence of neutrino oscillations observed in multitude of experiments confirms that neutrinos mix with each other and have non zero mass Super-Kamiokande:1998kpq . Neutrino oscillation phenomena can be parametrized in terms of six independent parameters, namely three mixing angles $(\theta_{13},\theta_{12},\theta_{23})$, one Dirac CP violating phase $(\delta_{\rm CP})$, and two mass squared differences $(\Delta m_{21}^{2}\,,\Delta m_{31}^{2})$. Although we have very precise values of the mixing angles and the absolute value of the mass squared differences, there are still some unknowns such as the octant of $\theta_{23}$, $\delta_{\rm CP}$ and the sign of $\Delta m_{31}^{2}$. There are two possible mass ordering of the neutrino mass spectrum: normal mass ordering (NO) $m_{1}<m_{2}<<m_{3}$ and inverted mass ordering (IO) $m_{3}<<m_{1}<m_{2}$ depending on the sign of $\Delta m_{31}^{2}$.

Neutrino oscillation experiments are sensitive only to the mass squared differences. They can not provide any information regarding the absolute mass scale of neutrinos which is one of the most sought after questions in particle physics today. Knowledge of the absolute mass scale of neutrinos is of great importance not only in particle physics but also in understanding the large scale structure of our universe. Neutrinos possess very tiny mass and unlike all other fermions in the Standard Model (SM), they do not seem to get their mass through Higgs mechanism. Hence, it may, in principle, help shape our understanding of the origin of particle mass which is still one of the most fundamental questions of particle physics. Unlike all other fermions in the SM, we observe only left handed neutrinos and right handed anti-neutrinos. We have not found any right handed neutrino and left handed anti-neutrino so far in experiments. This brings us to the next relevant question whether neutrino is a Dirac particle or a Majorana particle. Neutrino interactions could violate CP as well which will be crucial in explaining the matter antimatter asymmetry in the universe. Moreover, there could be additional sterile neutrinos.

There are several experimental efforts to find the absolute mass of the neutrino. The $\beta$ decay experiment performed at KATRIN can, in principle, measure the effective electron anti-neutrino mass by studying the end point region of the $\beta$ decay spectrum. This is completely model independent determination, i.e, it depends neither on any cosmological models nor on the nature of neutrinos. At present, the improved upper bound on the effective electron anti-neutrino mass is reported to be $m_{\nu}<0.8\,{\rm eV}$ at $90\%$ confidence level. The KATRIN experiment will continue to take data over the next several years and it is expected that the mass sensitivity will reach up to $0.2\,{\rm eV}$. Future experiments like Project $8$ Project8:2022wqh , designed to measure the absolute mass scale of the neutrino, hopes to reach a goal of $40\,{\rm meV/c^{2}}$ neutrino mass sensitivity. Indirectly, one can have information on neutrino mass from cosmological observations. These cosmological observations are sensitive to the total neutrino mass and to the number of neutrino species. There are several results related to the total neutrino mass coming from various cosmological observations. Most of these indirect methods put a limit on the total neutrino mass to be less than $0.2\,{\rm eV}$. These results are, however, model dependent. They rely heavily on several cosmological assumptions. Current upper bound on the total neutrino mass is reported by the Planck satellite to be $\sum\,m_{i}<0.12\,{\rm eV}$ at $95\%$ confidence level combining BAO data with CMB data  Zhang:2020mox . If KATRIN’s mass sensitivity reach up to $0.2\,{\rm eV}$ in future, it can put severe constraint on several cosmological models. Rare double $\beta$ decay process with two anti neutrinos in the final state is allowed in the SM. In general double beta decay processes are powerful probes of beyond the SM physics. More specifically, if one observes neutrinoless double $\beta$ decay in experiments, it would confirm that neutrinos are Majorana in nature. One can determine the effective Majorana mass $M_{ee}$ by studying neutrinoless double beta decay. There exists several limits on the value of $M_{ee}$ using different isotopes. At present, the best limits are reported to be $m_{ee}<(0.079-0.18)\,{\rm eV}$, $m_{ee}<(0.075-0.35)\,{\rm eV}$ and $m_{ee}<(0.061-0.165)\,{\rm eV}$ GERDA:2013vls ; CUORE:2015hsf ; EXO-200:2014ofj ; KamLAND-Zen:2012mmx , respectively.

There are several theoretical efforts in explaining the origin of neutrino mass. The most natural way to understand neutrino mass is through seesaw mechanism. The neutrino mass matrix in the framework of Type-I seesaw mechanism is given by $M_{\nu}=-M_{D}M_{R}^{-1}M_{D}^{T}$, where $M_{D}$ is the Dirac neutrino mass matrix and $M_{R}$ is the Majorana mass matrix of the right handed neutrinos. Phenomenology of Majorana neutrino mass matrix has been studied extensively assuming zero textures of the neutrino mass matrix which may be realised from the zeros in $M_{D}$ or $M_{R}$. In literature, there have been phenomenological studies with texture one-zero  Lashin:2011dn ; Singh:2018tqu ; Gautam:2018izb , two-zeros Frampton:2002yf ; Xing:2002ta ; Xing:2002ap ; Lavoura:2004tu ; Dev:2006qe ; Kumar:2011vf ; Fritzsch:2011qv ; Ludl:2011vv ; Meloni:2012sx ; Grimus:2012zm ; Dev:2014dla ; Gautam:2016qyw ; Channey:2018cfj ; Singh:2019baq and more within the context of Pontecorvo Maki Nakagawa Sakata (PMNS), tribimaximal (TB) and trimaximal (TM) mixing matrix. Similarly, in Ref. Lashin:2007dm ; Lashin:2009yd ; Dev:2010if ; Dev:2010pf ; Tavartkiladze:2022pzf ; Araki:2012ip , the authors have studied the phenomenological implication of vanishing minors in the neutrino mass matrix. Moreover, in Ref. Liao:2013saa ; Dev:2013xca ; Wang:2013woa ; Whisnant:2015ovx ; Dev:2015lya ; Wang:2016tkm and Ref. Goswami:2008uv ; Dev:2009he ; Dev:2010pe ; Liu:2013oxa ; Dev:2013nua , the authors have explored the implication of cofactor zero and hybrid texture of the neutrino mass matrix. In case of zero textures, it is found that three or more zeros in neutrino mass matrix can not accommodate the current neutrino oscillation data. In Ref. Frampton:2002yf , the authors have found that out of fifteen possible two texture zeros cases only seven cases with PMNS mixing are allowed experimentally. Also out of fifteen possible two cofactors zero patterns only seven patterns are acceptable Lashin:2007dm . In case of TM mixing along with magic symmetry Gautam:2016qyw , the authors have found that only two cases are valid for two texture zero. TM mixing with one texture zero was studied in Ref. Gautam:2018izb and found that all six patterns are compatible with current neutrino oscillation data. For TB mixing along with the condition of texture zeros or vanishing minor Dev:2010pf only five patterns are allowed. In this work we study the implication of one vanishing minor in the neutrino mass matrix using trimaximal mixing.

Our paper is organized as follows. In Section. II, we briefly discuss the neutrino mass matrix using trimaximal mixing matrix. We find all the mixing parameters such as $\theta_{13}$, $\theta_{23}$, $\theta_{12}$ and the CP violating parameter $\delta_{CP}$ in terms of the unknown parameters $\theta$ and $\phi$ of the trimaximal mixing matrix. In Section. III, we describe the formalism of one vanishing minor in neutrino mass matrix and identify all the possible patterns of one vanishing minor. We provide all the detail numerical analysis and discussion of each pattern in Sections. IV and  V. The fine-tuning of neutrino mass matrix is presented in Section. VI. In Section. VII, we present the symmetry realization and conclude in Section. VIII.

The most widely studied lepton flavor mixing is TB mixing pattern Harrison:2002er ; Harrison:2002kp ; Xing:2002sw ; Harrison:2003aw introduced by Harrison, Perkins and Scott. TB mixing pattern provides remarkable agreement with the atmospheric and solar neutrino oscillation data. The TB mixing pattern is given by

The TB mixing matrix possesses two types of symmetries: $\mu-\tau$ symmetry and magic symmetry. Although TB mixing matrix correctly predicted the value of atmospheric mixing angle $\theta_{23}$ and the solar mixing angle $\theta_{12}$, it, however, failed to explain a non zero value of the reactor mixing angle $\theta_{13}$ that was experimentally confirmed by T2K T2K:2011ypd , MINOS MINOS:2011amj , Double Chooz DoubleChooz:2011ymz , Daya Bay DayaBay:2012fng and RENO RENO:2012mkc experiments. The possibility of an exact $\mu-\tau$ symmetry in the mass matrix was completely ruled out by a relatively large value of $\theta_{13}$. Modifications in the TB mixing pattern Kumar:2010qz ; He:2011gb ; Grimus:2008tt was made to accommodate the present data. The TM mixing matrix was constructed by multiplying the TB mixing matrix by an unitary matrix and can be written as

and

where $\theta$ and $\phi$ are two free parameters. The neutrino mass matrix corresponding to TM mixing matrix can be written as

where $M_{diag}={\rm diag}(m_{1},m_{2},m_{3})$ is the diagonal matrix containing three mass state, $V=U_{TM}P$ and $P$ is the phase matrix written as

Here $\alpha$ and $\beta$ are the two CP violating Majorana phases.

There are six independent minors corresponding to six independent elements in the neutrino mass matrix. We denote the minor corresponding to $ij^{th}$ element of $M_{ij}$ as $C_{ij}$. The six possible patterns of one minor zero in neutrino mass matrix are listed in Table. 1.

The condition for one vanishing minor can be written as

More specifically, we can write Eq. 19 in terms of a complex equation as

where

with $(i,j,k)$ as the cyclic permutation of $(1,2,3)$. Using Eq. 20, one can write the two mass ratios as

The value of $m_{1}$, $m_{2}$ and $m_{3}$ can be calculated using Eq. III and mass square difference $\Delta m_{21}^{2}$. That is

Similarly, the ratio of squared mass difference is defined as

where $\Delta m_{21}^{2}$ and $\Delta m_{32}^{2}$ represent solar and atmospheric mass squared difference, respectively. Value of $r=(2.950\pm 0.08)\times 10^{-2}$ is determined by using the measured values of $\Delta m_{21}^{2}$ and $\Delta m_{32}^{2}$ reported in Ref. Esteban:2020cvm .

For our numerical analysis, we use the measured values of the oscillation parameters reported in Ref. Esteban:2020cvm . For completeness, we report them in Table. 2.

We wish to find the value of the unknown parameters $\theta$ and $\phi$. It is evident from Eq. 7 and Eq. II.2 that the neutrino oscillation parameters $\theta_{12}$ and $\theta_{13}$ depend only on $\theta$. To find the best fit value of $\theta$, we perform a naive $\chi^{2}$ analysis. The relevant $\chi^{2}$ is defined as

where $\theta_{i}=(\theta_{12},\theta_{13})$. Here $\theta_{i}^{cal}$ represents the theoretical value of $\theta_{i}$ and $\theta_{i}^{exp}$ represents measured central value of $\theta_{i}$. The corresponding uncertainties in the measured value of $\theta_{i}$ is represented by $\sigma_{i}^{exp}$.

For the TM₁ mixing matrix, the best fit value of $\theta$ is obtained to be $14.96^{\circ}$. The corresponding best fit values of $\theta_{12}$ and $\theta_{13}$ are $34.33^{\circ}$ and $8.57^{\circ}$, respectively. The $3\sigma$ allowed range of $\theta$ is found to be $(14.26^{\circ}-15.64^{\circ})$. Using the allowed range of $\theta$, we obtain the allowed ranges of $\theta_{12}$ and $\theta_{23}$ to be $(34.25^{\circ}-34.42^{\circ})$ and $(32.11^{\circ}-57.88^{\circ})$, respectively. We show in Fig. 1(a) the correlation of $\theta_{13}$ and $\theta_{12}$ for the TM₁ mixing matrix. To see the variation of $\theta_{23}$ with $\phi$, we use the allowed range of $\theta$ and vary $\phi$ within its full range from $0^{\circ}$ to $360^{\circ}$. We show in Fig. 1(b) the variation of $\theta_{23}$ as a function of the unknown parameter $\phi$. We also obtain the best fit value of $\phi$ by using the measured best fit value of $\theta_{23}$. The best fit value is shown with a ’$*$’ mark in Fig. 1(b). The best fit values of $\phi$ corresponding to the best fit value of $\theta_{23}=49.2^{\circ}$ are $69.43^{\circ}$ and $290.57^{\circ}$, respectively. We show the variation of $J$ and $\delta$ as a function of $\phi$ in Fig. 1(c) and Fig. 1(d), respectively. It is observed that the Jarlskog rephasing invariant $J$ and the Dirac CP violating phase $\delta$ are restricted to two regions. The corresponding best fit values of $J$ and $\delta$ are $[-3.184\times 10^{-2},3.185\times 10^{-2}]$ and $[71.11^{\circ},288.98^{\circ}]$, respectively. We also obtain the $3\sigma$ allowed ranges of $J$ and $\delta$ to be $[0,\pm 3.53\times 10^{-2}]$ and $[(55.05,124.95)^{\circ},\,(235.05,304.95)^{\circ}]$, respectively.

For TM₂ mixing matrix, the best fit value of $\theta$ is obtained to be $10.50^{\circ}$. The corresponding best fit values of $\theta_{12}$ and $\theta_{13}$ are $35.72^{\circ}$ and $8.56^{\circ}$, respectively. The $3\sigma$ allowed range of $\theta$ is found to be $(10.03^{\circ}-10.99^{\circ})$. Using the allowed range of $\theta$, we obtain the allowed ranges of $\theta_{12}$ and $\theta_{23}$ to be $(35.68^{\circ}-35.76^{\circ})$ and $(39.50^{\circ}-51.40^{\circ})$, respectively. It should be noted that although the allowed range of $\theta_{12}$ obtained with TM₂ mixing matrix is consistent with the $3\sigma$ experimental range, the best fit value obtained for $\theta_{12}$, however, deviates from the experimental best fit value at more than $2\sigma$ significance. This is quite a generic feature of TM₂ mixing matrix because, by default, value of $\theta_{12}$ will be greater than or equal to the value obtained in case of TB mixing matrix. We show in Fig. 2(a) the correlation of $\theta_{13}$ and $\theta_{12}$ for the TM₂ mixing matrix.

To see the variation of $\theta_{23}$ with $\phi$, we use the allowed range of $\theta$ and vary $\phi$ within its full range from $0^{\circ}$ to $360^{\circ}$. We show in Fig. 2(b) the variation of $\theta_{23}$ as a function of the unknown parameter $\phi$. The best fit value is shown with a ’$*$’ mark in Fig. 2(b). We obtain the best fit value of $\phi$ by using the measured best fit value of $\theta_{23}$. The best fit values of $\phi$ corresponding to the best fit value of $\theta_{23}=49.2^{\circ}$ are $44.86^{\circ}$ and $315.19^{\circ}$, respectively. We get two best fit values of $\phi$ because $\theta_{23}$ is invariant under the transformation $\phi\to(2\pi-\phi)$ which is evident from Eq. 7 and Eq. II.2. We also show the variation of $J$ and $\delta$ as a function of $\phi$ in Fig. 2(c) and Fig. 2(d), respectively. It is observed that the Jarlskog rephasing invariant $J$ and the Dirac CP violating phase $\delta$ are restricted to two regions. The corresponding best fit values of $J$ and $\delta$ are $[2.37\times 10^{-2},-2.50\times 10^{-2}]$ and $[45.48^{\circ},314.44^{\circ}]$, respectively. We also obtain the $3\sigma$ allowed ranges of $J$ and $\delta$ to be $[0,\pm 3.60\times 10^{-2}]$ and $[(0,90)^{\circ},\,(270,360)^{\circ}]$, respectively.

In case of inverted mass ordering the $3\sigma$ allowed range of $\theta$ are found to be $(14.37^{\circ}-15.64^{\circ})$ and $(10.10^{\circ}-10.99^{\circ})$ for both TM₁ and TM₂ mixing matrix, respectively. Using the $3\sigma$ allowed range of $\theta$, we obtain the $3\sigma$ allowed ranges of $\theta_{12}$ to be $(34.25^{\circ}-34.41^{\circ})$ and $(35.68^{\circ}-35.76^{\circ})$, respectively for both TM₁ and TM₂ mixing matrix. The $3\sigma$ allowed ranges of $\theta_{23}$ to be $(32.12^{\circ}-57.87^{\circ})$ and $(38.60^{\circ}-51.39^{\circ})$, respectively for both TM₁ and TM₂ mixing matrix. It is to be noted that the mixing angles are almost similar for both the normal and inverted mass ordering. So for our later discussion we will use the values of mixing angles for the normal mass ordering reported in Ref. Esteban:2020cvm .

We wish to investigate the phenomenological implication of one vanishing minor in the neutrino mass matrix on the total neutrino mass, the effective Majorana mass term and the electron anti-neutrino mass. It is evident from Eq. III that neutrino mass $m_{i}$ depends on $\theta$, $\phi$, $\alpha$, $\beta$ and the mass squared difference $\Delta m_{21}^{2}$. We use the best fit value and the $3\sigma$ allowed range of $\theta$ and $\phi$ of section. IV that are determined by the measured values of the mixing angles $\theta_{13}$, $\theta_{12}$ and $\theta_{23}$. The two unknown Majorana phases $\alpha$ and $\beta$ are varied within their full range from $0^{\circ}$ to $360^{\circ}$. Moreover, we use the $3\sigma$ allowed ranges of $\Delta m_{21}^{2}$ and $r$ to constrain the values of the neutrino masses. Now we proceed to analyse all the six patterns of one vanishing minor one by one.