Implications of texture zeros for a variant of tribimaximal Mixing

TL;DR

This paper investigates how specific zero patterns in the neutrino mass matrix, combined with tribimaximal mixing, lead to testable predictions for neutrino experiments, narrowing down viable models.

Contribution

It identifies two viable texture zero patterns compatible with data, linking them to tribimaximal mixing and providing testable predictions for future experiments.

Findings

Two texture zero patterns are compatible with experimental data.

Predictions for neutrino observables are derived from these textures.

The study constrains neutrino mass models based on texture zeros.

Abstract

We study the phenomenological implications of the presence of two texture zeros in the neutrino mass matrix assuming that the neutrino mixing matrix has its first column identical to that of the tribimaximal mixing matrix. Only two patterns of this kind are compatible with the experimental data. These textures have definite predictions for the neutrino observables that are testable in future neutrino experiments.

Tables

Table 1. Table 1: Seven allowed mass matrices with two zeros classified into three classes.

Type	Constraining Equations
A1	$M_{ee}=0$, $M_{e\mu }=0$
A2	$M_{ee}=0$, $M_{e\tau }=0$
B1	$M_{e\tau }=0$, $M_{\mu \mu }=0$
B2	$M_{e\mu }=0$, $M_{\tau \tau }=0$
B3	$M_{e\mu }=0$, $M_{\mu \mu }=0$
B4	$M_{e\tau }=0$, $M_{\tau \tau }=0$
C	$M_{\mu \mu }=0$, $M_{\tau \tau }=0$

Table 2. Table 2: Allowed 3 σ 𝜎 \sigma ranges of θ 23 subscript 𝜃 23 \theta_{23} and δ 𝛿 \delta for patterns A 1 and A 2 . Solution - I and Solution - II are two allowed solutions for δ 𝛿 \delta .

Type	${\theta }_{23}$	$\delta$
		Solution - I	Solution - II
A1	41.18∘ - 44.02∘	94.6∘ - 106.5∘	253.5∘ - 265.5∘
A2	45.98∘ - 48.82∘	73.5∘ - 85.5∘	274.6∘ - 286.5∘

Equations

$$U_{\text{TBM}}=\left(\begin{array}[]{ccc}\sqrt{\frac{2}{3}}&\frac{1}{\sqrt{3}}&0\\ -\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}&-\frac{1}{\sqrt{2}}\\ \end{array}\right).$$

$$U_{\text{TM}_{1}}=\left(\begin{array}[]{ccc}\frac{2}{\sqrt{3}}&\frac{1}{\sqrt{3}}\cos\theta&\frac{1}{\sqrt{3}}\sin\theta\\ -\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}\cos\theta-\frac{e^{i\phi}\sin\theta}{\sqrt{2}}&\frac{1}{\sqrt{3}}\sin\theta+\frac{e^{i\phi}\cos\theta}{\sqrt{2}}\\ -\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}\cos\theta+\frac{e^{i\phi}\sin\theta}{\sqrt{2}}&\frac{1}{\sqrt{3}}\sin\theta-\frac{e^{i\phi}\cos\theta}{\sqrt{2}}\end{array}\right).$$

$$U_{\text{TM}_{2}}=\left(\begin{array}[]{ccc}\sqrt{\frac{2}{3}}\cos\theta&\frac{1}{\sqrt{3}}&\sqrt{\frac{2}{3}}\sin\theta\\ -\frac{\cos\theta}{\sqrt{6}}+\frac{e^{-i\phi}\sin\theta}{\sqrt{2}}&\frac{1}{\sqrt{3}}&-\frac{\sin\theta}{\sqrt{6}}-\frac{e^{-i\phi}\cos\theta}{\sqrt{2}}\\ -\frac{\cos\theta}{\sqrt{6}}-\frac{e^{-i\phi}\sin\theta}{\sqrt{2}}&\frac{1}{\sqrt{3}}&-\frac{\sin\theta}{\sqrt{6}}+\frac{e^{-i\phi}\cos\theta}{\sqrt{2}}\end{array}\right).$$

$$U_{\text{TM}_{3}}=\left(\begin{array}[]{ccc}\cos\theta&\sin\theta&0\\ -\frac{e^{-i\phi}\sin\theta}{\sqrt{2}}&\frac{e^{-i\phi}\cos\theta}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\ -\frac{e^{-i\phi}\sin\theta}{\sqrt{2}}&\frac{e^{-i\phi}\cos\theta}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\end{array}\right).$$

$$M_{\text{TM}_{1}}=\left(\begin{array}[]{ccc}2a&2b&2c\\ 2b&b-c+d&2a+2b-d\\ 2c&2a+2b-d&-3b+3c+d\end{array}\right).$$

$$M^{\text{A}_{1}}=\left(\begin{array}[]{ccc}0&0&2c\\ 0&-2c+\Delta&c-\Delta\\ 2c&c-\Delta&2c+\Delta\end{array}\right)$$

$$M^{\text{A}_{2}}=\left(\begin{array}[]{ccc}0&2b&0\\ 2b&2b+\Delta&b-\Delta\\ 0&b-\Delta&-2b+\Delta\end{array}\right),$$

$$M^{\text{B}_{1}}=\left(\begin{array}[]{ccc}2a&2b&0\\ 2b&0&2a+3b\\ 0&2a+3b&-4b\end{array}\right),$$

$$M^{\text{B}_{2}}=\left(\begin{array}[]{ccc}2a&0&2c\\ 0&-4c&2a+3c\\ 2c&2a+3c&0\end{array}\right),$$

$$M^{\text{B}_{3}}=\left(\begin{array}[]{ccc}2a&0&2c\\ 0&0&2a-c\\ 2c&2a-c&4c\end{array}\right),$$

$$M^{\text{B}_{4}}=\left(\begin{array}[]{ccc}2a&2b&0\\ 2b&4b&2a-b\\ 0&2a-b&0\end{array}\right).$$

$$M^{C}=\left(\begin{array}[]{ccc}2a&2b&2b\\ 2b&0&2a+2b\\ 2b&2a+2b&0\end{array}\right).$$

$${\theta }_{23}\to \frac{\pi }{2}-{\theta }_{23},\delta \to \pi -\delta$$

$$U^{T}MU=M_{\text{diag}}$$

$$M_{\text{diag}}=\left(\begin{array}[]{ccc}m_{1}&0&0\\ 0&e^{2i\alpha}m_{2}&0\\ 0&0&e^{2i\beta}m_{3}\end{array}\right).$$

$$s_{12}^2=\frac{|U_{12}{|}^2}{1-|U_{13}{|}^2},s_{23}^2=\frac{|U_{23}{|}^2}{1-|U_{13}{|}^2}\text{ and }{s}_{13}^2=|U_{13}{|}^2,$$

$$s_{12}^2=1-\frac{2}{3-{\sin }^2\theta },$$

$$s_{23}^2=\frac{1}{2}\left(1+\frac{\sqrt{6}\sin 2\theta \cos \varphi }{3-{\sin }^2\theta }\right),$$

$$s_{13}^2=\frac{1}{3}{\sin }^2\theta .$$

$$J=Im(U_{11}{U}_{12}^{\ast }{U}_{21}^{\ast }{U}_{22})$$

$$J=s_{12}{s}_{23}{s}_{13}{c}_{12}{c}_{23}{c}_{13}^2\sin \delta .$$

$$J=\frac{1}{6\sqrt{6}}\sin 2\theta \sin \varphi .$$

$${\cot }^2\delta ={\cot }^2\varphi -\frac{6{\sin }^{2}2\theta {\cot }^2\varphi }{(3-{\sin }^2\theta {)}^2}.$$

$$M_{\nu }=U^{\ast }{M}_{\text{diag}}{U}^{†}$$

$$\frac{m_1}{m_2}=\frac{\sin 2(\alpha -\beta ){\cos }^2\theta }{2\sin 2\beta }$$

$$\frac{m_2}{m_3}=-\frac{\sin 2\beta {\tan }^2\theta }{\sin 2\alpha }.$$

$$\cot 2\alpha =\cot 2\beta +\frac{m_1}{m_3}\csc 2\beta {\cot }^2\theta .$$

$$\frac{\sin 2(\alpha -\beta )}{2m_1}-\frac{\sin 2\beta }{m_2}+\frac{\sin 2\alpha }{m_3}=0.$$

$$r=\frac{-{\sin }^{2}2(\alpha -\beta )+4{\sec }^4\theta {\sin }^{2}2\beta }{-{\sin }^{2}2(\alpha -\beta )+4{\csc }^4\theta {\sin }^{2}2\beta }.$$

$${\sin }^2\theta =-\frac{\sin 2\alpha \sin (2\beta -\varphi )}{\sin 2(\alpha -\beta )\sin \varphi }.$$

Full text

In the recent years, considerable efforts have been put towards determining the structure of the neutrino mass matrix ($M_{\nu}$)reviews . The non-zero value of the reactor mixing angle $\theta_{13}$, recently determined in various neutrino oscillation experiments th13 , has called many neutrino mass models predicting $\theta_{13}=0$ into question. The models based upon the Tribimaximal (TBM) hps mixing, that predicted the reactor, atmospheric and solar mixing angles as $\theta_{13}=0$, $\theta_{23}=\frac{\pi}{2}$, and $\theta_{12}=\arctan(1/\sqrt{2})$, respectively, need modifications in the light of a non-zero $\theta_{13}$. The TBM mixing matrix is given as

[TABLE]

Many ways have been proposed to modify the TBM ansatz to accommodate a non-zero $\theta_{13}$. A simple possibility is to keep one column of TBM mixing matrix unchanged while modifying its other two columns within unitarity constraints partial . This gives rise to three Trimaximal (TM) mixing patterns, viz., TM1, TM2, and TM3, that have their first, second and third columns identical to TBM matrix, respectively partial . These three mixing schemes contain TBM mixing as a special case that enlarges the symmetry of these mixing patterns. Hence, they could have been named TBM1, TBM2, and TBM3.

TM1 mixing is given as

[TABLE]

The mixing scheme reduces to the TBM scheme in the special case $\theta=0$ and $\phi=0$.

TM2 mixing is given as

[TABLE]

The mixing scheme reduces to the TBM scheme in the special case $\theta=0$ and $\phi=0$. This mixing scheme corresponds to the magic symmetry.

TM3 mixing is given as

[TABLE]

The mixing scheme reduces to the TBM scheme in the special case $\theta=\arctan(1/\sqrt{2})$ and $\phi=0$. This mixing scheme is equivalent to $\mu-\tau$ symmetry.

Another simple assumption that can accommodate a non-zero $\theta_{13}$ is the presence of texture zeros in the neutrino mass matrix fgm ; xingtz ; tz . Texture zeros induce relations between mixing matrix elements and neutrino masses. Considering neutrinos to be Majorana fermions and working in a basis where the charged lepton mass matrix $M_{l}$ is diagonal, there are in total fifteen different patterns of two texture zeros in the neutrino mass matrices. Out of these fifteen possible patterns, only seven can satisfy the present neutrino oscillation data fgm ; xingtz . These seven patterns are classified in three classes A, B and C corresponding to the normal, quasi-degenerate and inverted mass hierarchies of neutrinos [Table 1].

The current neutrino data are consistent with the possibility of keeping first or second column of the mixing matrix unmodified (TM1 or TM2 mixing) while modifying other columns within unitarity constraints. The experimental data is also consistent with the presence of two texture zeros in the neutrino mass matrix. If we combine both approaches together by having texture zeros in a mass matrix corresponding to TM1 or TM2 mixing, we are bound to get very predictive neutrino mass matrices. In an earlier work 2tztm , we studied the implications of two texture zeros in a magic mass matrix giving TM2 mixing. In the present work, we study the phenomenological implications of the presence of two texture zeros in a neutrino mass matrix giving TM1 mixing.

A mass matrix giving TM1 mixing can be parameterized as

[TABLE]

We can obtain the form of the neutrino mass matrix giving TM1 mixing for the seven patterns of two texture zeros by substituting the respective constraints from Table 1 in Eq. (5).

The neutrino mass matrix of type A1 giving TM1 mixing is given as

[TABLE]

where $\Delta=d+b$. Similarly, the neutrino mass matrix of type A2 giving TM1 mixing is

[TABLE]

where $\Delta=d-c$.

The four mass matrices giving TM1 mixing for class B are

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

We will show that all these mass matrices of type class B giving TM1 mixing are not allowed by the experimental data.

The mass matrix with TM1 mixing for the class C is

[TABLE]

This mass matrix has $\mu-\tau$ symmetry and implies $\theta_{13}=0$. Hence, it is not allowed.

The phenomenology of patterns A1 and A2 is related: one can obtain the predictions for A2 by making the transformations tz ; xingtz

[TABLE]

on the predictions of A1. Hence, we study the phenomenological implications for pattern A1 only.

Any mass matrix $M$ giving TM1 mixing can be diagonalized by a mixing matrix $U=U_{\text{TM}_{1}}$ given in Eq. (2) using the relation

[TABLE]

where $M_{\text{diag}}$ is the diagonal mass matrix given as

[TABLE]

Here, $m_{1}$, $m_{2}$, and $m_{3}$ are the neutrino masses and $\alpha$ and $\beta$ are the two Majorana phases.

Once the mixing matrix $U$ is known, the mixing angles can be calculated using the relations:

[TABLE]

where $s_{ij}=\sin\theta_{ij}$ and $c_{ij}=\cos\theta_{ij}$. For our case $U=U_{\text{TM}_{1}}$, the above relations give

[TABLE]

[TABLE]

and

[TABLE]

We see from Eq. (17) that $\theta_{12}$ is smaller than its TBM value $s_{12}^{2}=1/3$. In contrast, the value of $\theta_{12}$ is larger than the TBM value for TM2 mixing. Since the experimental value of $\theta_{12}$ is towards the lower side of the TBM value, TM1 mixing is more appealing than TM2 mixing.

The CP violating phase $\delta$ can be calculated from the Jarlskog rephasing invariant measure of CP violation jarlskog

[TABLE]

using the relation

[TABLE]

Substituting the elements of the TM1 mixing matrix in Eq. (20), we obtain

[TABLE]

From Eqs. (21) and (22), we get

[TABLE]

We reconstruct the neutrino mass matrix for TM1 mixing using the relation:

[TABLE]

where $U=U_{\text{TM}_{1}}$. To obtain the predictions for the neutrino mass matrix of the type A1 given by Eq. (6), we have to solve the two complex equations: $M_{\nu_{11}}=0$ and $M_{\nu_{12}}=0$. Solving the equation $M_{\nu_{11}}=0$, we get

[TABLE]

and

[TABLE]

Using these two equations, we evaluate $m_{1}/m_{3}$ and invert the resulting relation to obtain

[TABLE]

We note that the presence of a zero at (1,1) entry in a mass matrix with TM1 mixing, through Eqs. (25) and (26), implies a beautiful sum-rule on neutrino masses:

[TABLE]

The texture zero at (1,1) entry in a mass matrix with TM1 mixing also gives a prediction for the ratio $r=\Delta m^{2}_{21}/\Delta m^{2}_{31}$. From Eqs. (25) and (26), we obtain

[TABLE]

We solve the second equation $M_{\nu_{12}}=0$ by equating its real and imaginary parts to zero. We eliminate $m_{1}$ and $m_{2}$ from the resulting equations using Eqs. (25) and (26). Equating the imaginary part of $M_{\nu_{12}}$ to zero, we obtain

[TABLE]

We get a quadratic equation in $\tan^{2}\theta$ on equating the real part of $M_{\nu_{12}}$ to zero:

[TABLE]

Solving this equation by substituting $\alpha$ from Eq. (27), we obtain

[TABLE]

The value of $\theta$ calculated in Eqs. (30) and (32) must be identical. This requirement gives

[TABLE]

Equations (25), (26), (32) and (33) are the four predictions for the neutrino mass matrix of type A1. We can express these four predictions as expressions for $\alpha$, $\beta$, $\frac{m_{1}}{m_{2}}$, $\frac{m_{2}}{m_{3}}$ as functions of $\theta$ and $\phi$. Inversion of Eq. (32) gives

[TABLE]

Substituting $\beta$ from Eq. (34) in Eq. (33) gives

[TABLE]

Equations(25) and (26) after substituting values of $\alpha$ and $\beta$ give

[TABLE]

and

[TABLE]

We can use these ratios to calculate $r=\frac{\Delta m_{21}^{2}}{\Delta m_{31}^{2}}$ as a function of $\theta$ and $\phi$ or we can calculate it directly from Eq. (29). We obtain

[TABLE]

For TM2 mixing, we have $r=\tan^{2}\theta$ 2tztm . In contrast, $r$ is function of both $\theta$ and $\phi$ for TM1 mixing. By demanding that $r$ satisfies its experimental value, one can calculate experimentally allowed values of $\phi$.

Once the experimentally allowed values of $\theta$ and $\phi$ are known, the observables $\theta_{12},\theta_{23},\theta_{13}$ and $\delta$ can be calculated from Eqs. (17), (18), (19) and (23) as they are functions of $\theta$ and $\phi$.

The experimentally allowed value of $\theta$ can be calculated from Eq. (19) using the experimental value of $\sin^{2}\theta_{13}=0.0216\pm 0.00075$ data . We get $\theta=14.78\pm 0.26$ degrees. We could have also used the experimental value of $\sin^{2}\theta_{12}=0.306\pm 0.012$ data to constrain $\theta$. However, this gives a larger range of $\theta$. We plot $r$ as a function of $\phi$ using Eq. (38) for $\theta=14.78\pm 0.26$ degrees in Fig. 1. From the experimental values $\Delta m_{21}^{2}=(7.50\pm 0.19)\times 10^{-5}$ eV2 and $\Delta m_{31}^{2}=(2.524\pm 0.04)\times 10^{-3}$ eV2 data , we get the experimental value $r=0.0297\pm 0.0003$. Here, all the experimental errors are at one standard deviation. We find that the predicted and experimental values of $r$ are consistent only for two regions of $\phi$ depicted in Fig. 1. In this way, we can constrain both $\theta$ and $\phi$. Then, all other observables can be obtained as they are functions of $\theta$ and $\phi$.

A better approach is to constrain $\theta$ and $\phi$ by minimizing the $\chi^{2}$-function

[TABLE]

where the variables $v_{i}=(\sin^{2}\theta_{12},\sin^{2}\theta_{13},r)$; $v_{i}^{exp}$ are experimental values of $v_{i}$; and $\sigma_{i}^{exp}$ are the standard deviations in the experimental values of $v_{i}$. The $\chi^{2}(\theta,\phi)$ is minimum at $\theta=14.8$ degrees and $\phi=103.2$ degrees or $\phi=256.8$ degrees. The minimum value is $\chi^{2}_{min}=1.8$. The contours of $\Delta\chi^{2}=\chi^{2}-\chi_{min}$ corresponding to 1$\sigma$, 2$\sigma$, and 3$\sigma$ confidence level are shown in Fig. 2. The predictions for $\theta_{23}$ and $\delta$ for TM1 mixing in pattern A1 have been shown in Fig. 3 at various confidence levels. The predictions for TM1 mixing in pattern A2 can be obtained by using the transformations given in Eq. (13). We find that $\delta$ is predicted either around 100∘ or around 260∘ with a spread of around 10∘. Table 2 shows the 3$\sigma$ ranges of $\delta$ and $\theta_{23}$ for these two solutions obtained from our analysis. The mixing angle $\theta_{23}$ lies below (above) 45∘ for pattern A1 (A2).

Recently, long baseline neutrino oscillation experiments like MINOS and T2K lbl are showing a preference for the CP violating phase $\delta$ to be around 270∘. In particular, a recent global analysis in Ref. data rules out $\delta$ from about 55∘ to 120∘ at 3$\sigma$ CL for inverted mass ordering. A certain portion of $\delta$ is also ruled out if $\theta_{23}<$ 45∘ for the normal mass ordering (see Fig. 11 in Ref. data ). In our $\chi^{2}$ analysis, we do not put any experimental constraints on $\delta$ and $\theta_{23}$ as these parameters are not precisely measured and their distributions are not Gaussian. If we take into consideration the limits on $\delta$ as given in Ref. data , the Solution - I in Table 2 for pattern A1 is ruled out and only Solution - II, where $\delta$ lies around 260∘, remains compatible. Since the exclusion region of $\delta$ with respect to $\theta_{23}$ for normal mass spectrum (Fig. 11 of Ref. data ) is not symmetric around $\theta_{23}=$ 45∘, both solutions are still allowed for pattern A2.

The predictions for Majorana phases $\alpha$ and $\beta$ are shown in Fig. 4. We also depict the predictions for the effective electron neutrino mass for $\beta$-decay $m_{\beta}^{2}=m_{1}^{2}|U_{e1}|^{2}+m_{2}^{2}|U_{e2}|^{2}+m_{3}^{2}|U_{e3}|^{2}$ and the sum of neutrino masses $\sum m_{i}=m_{1}+m_{2}+m_{3}$ in Fig. 5. Since the (1,1) element of the neutrino mass matrix vanishes for patterns A1 and A2, this leads to a vanishing effective Majorana neutrino mass ($|M_{ee}|=|m_{1}U_{e1}^{2}+m_{2}U_{e2}^{2}+m_{3}U_{e3}^{2}|$) for these patterns.

The neutrino mass matrix of type B1 has zeros at $(1,3)$ and $(2,2)$ entries. This implies following expression of ratio $r$ in the presence of TM1 mixing:

[TABLE]

We can express $\theta$ in terms of $\theta_{13}$ using Eq. (19) in the above relation expressing $r$ as a function of $\theta$ (Fig. 6). It is clear that we cannot have both $r$ and $\theta_{13}$ in their experimentally allowed ranges simultaneously. Hence, this pattern is inconsistent with the experimental data when combined with TM1 mixing. The neutrino mass matrix of type B2 is related to the neutrino mass matrix of type B1 by a $\mu$-$\tau$ exchange tz ; xingtz and has identical predictions for $r$ and $\theta_{13}$. Hence, neutrino mass matrix of type B2 with TM1 mixing is also incompatible with the recent experimental data.

The neutrino mass matrix of type B3 has zeros at $(1,2)$ and $(2,2)$ entries. The expression of ratio $r$ in the presence of TM1 mixing is given by

[TABLE]

In case of pattern B3, the parameter $r$ always remains larger than 0.4 whereas the experimental range of this parameter lies well below the value 0.4 (see Fig. 6). Thus pattern B3 is also incompatible with the experimental data. Since pattern B3 is related to pattern B4 by $\mu$-$\tau$ exchange symmetry, the pattern B4 is also incompatible with the experimental data when combined with TM1 mixing.

In conclusion, we have studied the phenomenological implications of two texture zeros in the presence of TM1 mixing. There are seven allowed patterns for the presence of two texture zeros in the neutrino mass matrix. The presence of TM1 mixing rules out five out of the seven patterns of two texture zeros. The neutrino mass matrix having two texture zeros and TM1 mixing simultaneously can only belong to patterns A1 and A2. The Dirac CP violating phase $\delta$ is restricted to two narrow regions around $100^{\circ}$ and $260^{\circ}$ for these patterns. For TM1 mixing, $\theta_{12}$ is smaller than its TBM value and moves towards its best fit value with the increase in $\theta$. The imposition of TM1 mixing on two zeros make these classes very predictive and these predictions can be tested in future neutrino oscillation experiments.