A study on a minimally broken residual TBM-Klein symmetry with its implications on flavoured leptogenesis and ultra high energy neutrino flux ratios

The structure of the leptonic mixing matrix $U_{\rm PMNS}$ has always been the center of attraction in the flavour model building landscape. Until the experimental discovery of a nonvanishing value of the reactor mixing angle $\theta_{13}$[1, 2], it was the paradigm of Tri-BiMaximal (TBM) Ansatz of $U_{\rm PMNS}$[3], that dominated almost for a decade[4] with the prediction $\theta_{13}=0$, particularly in the approaches of model building with the discrete non-Abelian symmetries such as $A_{4}$, $S_{4}$ etc.[5]. Now the neutrino mixing parameters, particularly two mass squared differences and three mixing angles have entered in the precision era. Thus, as far as the TBM mixing is concerned, it has been rendered outdated at least by the nonvanishing value of $\theta_{13}$. However, due to its other two predictions, $\tan\theta_{23}=1$ and $\sin\theta_{12}=1/\sqrt{3}$ which are still close to their respective experimental best-fit values, the TBM mixing cannot be just overthrown from the way in search for an viable flavour model of neutrino masses and mixing. Several theories for the modification of TBM mixing in terms of perturbations to the TBM mass matrix have been proposed [6, 7]. A brief recall of few of the existing theories which deal with schemes of the modified TBM mixing would be worthwhile in our context. Ref.[6] has discussed the consequences of perturbation to the effective $M_{\nu}^{TBM}$ but with less emphasis on the high energy symmetries. On the other hand, in some of the works in Ref.[7], a high energy symmetry group $A_{4}$ is perturbed softly such that the effective residual symmetries are unable to generate the exact TBM mixing. Alternative moderated versions, such as TM1[8, 9], TM2[8, 10] mixing with an additional prediction on the Dirac CP phase $\delta$, have also been considered to comply with the existing neutrino oscillation data. Elaborate descriptions of direct models mainly focusing on TM1 and TM2 mixings have been given in references [11] and [12].

Besides all these, there exists a residual symmetry approach[13, 14, 15, 16, 17]. In [13, 14, 15] it has been shown that a $3\times 3$ neutrino Majorana mass matrix with nondegenerate eigenvalues always enjoys a $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ residual Klein symmetry accompanied with a diagonal charged lepton mass matrix which is further protected by a $\mathbb{Z}_{n}$ symmetry for $n>2$. Thus, given a neutrino mixing matrix, one can always construct the corresponding residual Klein symmetry generators for each of the $\mathbb{Z}_{2}$ symmetries. For a mixing of TBM kind, these generators are found to be identical to the generators of the $S_{4}$ group in a three dimensional irreducible representation[15]. The diagonal $\mathbb{Z}_{3}$ type symmetry acts as the residual symmetry in the charged lepton sector while the other two generate a TBM-Klein symmetry with one of them being the $\mu\tau$ interchange symmetry[18]. In[19], the consequences of a Scaling-Klein symmetry have also been worked out.

It is clear that the vanishing value of $\theta_{13}$ in TBM mixing is caused due to the existence of the $\mu\tau$ interchange symmetry as one of the TBM-Klein symmetry generators. Thus to be consistent with the oscillation data, one has to relax the constraints arising from the exact $\mu\tau$ symmetry. One way is to consider a $\mu\tau$ flavoured nonstandard CP symmetry (${\rm CP}^{\mu\tau}$)[20] instead of an exact $\mu\tau$ symmetry with the other TBM-Klein generator being completely broken. Introduction of such a symmetry leads to a co-bimaximal ($\theta_{23}=\pi/4$, $\cos\delta=0$) mixing[21]. This approach has drawn a lot of attention[22] after the recent hint from T2K about a maximal Dirac CP violation[23]. Following this approach, alternatives to TBM mixing have been proposed recently in[24] and [25]. In both the papers, the $\mu\tau$ interchange symmetry has been replaced by a $\mu\tau$ CP symmetry keeping the remaining generator of the TBM-Klein symmetry intact. This further makes the ${\rm CP}^{\mu\tau}$ more predictive with the added predictions of the unbroken TBM generator. In our present work we follow this direction, i.e., we keep a generator of the residual TBM-Klein symmetry unbroken and study modifications of the $\mu\tau$ interchange symmetry. However, instead of replacing the $\mu\tau$ interchange symmetry by ${\rm CP}^{\mu\tau}$, we have shown how minimal the breaking of the former could be to be consistent with the recent global fit neutrino oscillation data[26] or in other words, we have performed a study on the goodness of the $\mu\tau$ symmetry while keeping the other $\mathbb{Z}_{2}$ of the TBM-Klein symmetry unbroken. Unlike in [24, 25], here a nonmaximal value of $\theta_{23}$ is also allowed. We show that among the two relevant breaking parameters, one should always be large (more than 0.45). However, to be consistent with the other global fit data, whilst a maximality or a near maximality in $\theta_{23}$ requires large breaking (more than 0.35) in the other, a range $49.4^{\circ}-53^{\circ}$ or $38^{\circ}-40^{\circ}$ could be probed if we allow the same breaking parameter up to 0.25 (to be consistent with neutrino oscillation global fit data, at least $16\%$ breaking of the $\mu\tau$ symmetry is required in our scenario). Thus our model could be tested shortly in the experiments such as NO$\nu$A[27]. We have studied the breaking of the $\mu\tau$ interchange symmetry from the Lagrangian level of the Type-I seesaw. This in turn has allowed us to explore a scenario of leptogenesis with quasi-degenerate heavy RH neutrinos and to work out the consequences pertaining to a successful leptogenesis in this scheme.

Without loss of any generality we choose to work in the diagonal basis of charged leptons where the right handed neutrino mass matrix is also diagonal unless it is perturbed. The Lagrangian for the neutrino mass terms (Dirac+Majorana) is denoted as

$\displaystyle-\mathcal{L}_{mass}^{\nu,N}=\overline{\not{L}}_{L\alpha}(m_{D})_{l\alpha}N_{Rl}+\frac{1}{2}\overline{N_{Rl}^{C}}(M_{R})_{l}\delta_{lm}N_{Rm}+{\rm h.c}.,$

(1.1)

where $\not{L}_{L\alpha}=\begin{pmatrix}\nu_{L\alpha}&e_{L\alpha}\end{pmatrix}^{T}$ is the SM lepton doublet of flavour $\alpha$. The effective light neutrino mass matrix is then given by the well known Type-I seesaw formula

$\displaystyle M_{\nu}=-m_{D}M_{R}^{-1}m_{D}^{T}.$

(1.2)

A unitary matrix $U$ diagonalizes $M_{\nu}$ in (1.2) as

$$U^{T}M_{\nu}U=M_{\nu}^{d}\equiv diag\hskip 2.84526pt(m_{1},m_{2},m_{3}),$$

(1.3)

where $m_{i}\hskip 2.84526pt(i=1,2,3)$ are real positive mass eigenvalues of light neutrinos. Since $M_{\ell}$ is diagonal, $U$ is simply equivalent to the leptonic ${\rm PMNS}$ mixing matrix $U_{\rm PMNS}$:

$\displaystyle U=P_{\phi}U_{\rm PMNS}\equiv P_{\phi}\begin{pmatrix}c_{12}c_{13}&e^{i\frac{\alpha}{2}}s_{12}c_{13}&s_{13}e^{-i(\delta-\frac{\beta}{2})}\\ -s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}&e^{i\frac{\alpha}{2}}(c_{12}c_{23}-s_{12}s_{13}s_{23}e^{i\delta})&c_{13}s_{23}e^{i\frac{\beta}{2}}\\ s_{12}s_{23}-c_{12}s_{13}c_{23}e^{i\delta}&e^{i\frac{\alpha}{2}}(-c_{12}s_{23}-s_{12}s_{13}c_{23}e^{i\delta})&c_{13}c_{23}e^{i\frac{\beta}{2}}\end{pmatrix},$

(1.4)

where $P_{\phi}={\rm diag}~(e^{i\phi_{1}},~e^{i\phi_{2}},~e^{i\phi_{3}})$ is an unphysical diagonal phase matrix and $c_{ij}\equiv\cos\theta_{ij}$, $s_{ij}\equiv\sin\theta_{ij}$ with the mixing angles $\theta_{ij}=[0,\pi/2]$. Here we have followed the PDG convention[28] but denote our Majorana phases by $\alpha$ and $\beta$. CP-violation enters in the leptonic sector through nontrivial values of the Dirac phase $\delta$ and Majorana phases $\alpha,\beta$ with $\delta,\alpha,\beta=[0,2\pi]$.

A horizontal symmetry $G_{i}$ of a neutrino mass matrix is realized through the invariance equation

$\displaystyle G_{i}^{T}M_{\nu}G_{i}=M_{\nu},$

(2.1)

where $G_{i}$ is an unitary matrix in the neutrino flavour space. Now Eq.(2.1) and (1.3) together imply that we can define a new unitary matrix $V_{i}=G_{i}U$ such that it also diagonalizes $M_{\nu}$. The matrix $V$ should then be equal to $Ud_{i}$:

$\displaystyle G_{i}U=Ud_{i}\equiv U^{\dagger}G_{i}U=d_{i}$

(2.2)

with $d_{i}$ being a diagonal rephasing matrix. For neutrinos of Majorana type, $d_{i}^{2}=1$. Therefore, $d_{i}$ can only have entries $\pm 1$. Thus there are now eight possible structures of $d_{i}$ two of which are a simple unit matrix and its negative. The remaining six can be considered as three different pairs, where the two matrices of a pair are identical to each other apart from an overall relative negative sign. Finally, among these three (pairs) matrices, only two are independent as each $d_{i}$ always satisfies the relation $d_{i}=d_{j}d_{k}$, where $i,j$ and $k$ can take value 1, 2 and 3. Now $d_{i}^{2}=1$ implies each $d_{i}$ define a $\mathbb{Z}_{2}$ symmetry. Therefore, on account of the relation in (2.2), each $G_{i}~(G_{i}^{2}=1,~{\rm det}\hskip 2.84526pt(G_{i})=\pm 1)$ also represents $\mathbb{Z}_{2}$ symmetry and generates the residual $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry (Klein Symmetry) in the neutrino mass term of the Lagrangian. We can now choose the two independent $d_{i}$ matrices as $d_{2}={\rm diag}\hskip 2.84526pt(-1,1,-1)$ and $d_{3}={\rm diag}\hskip 2.84526pt(-1,-1,1)$ for ${\rm det}\hskip 2.84526pt(G_{i})=1$. For ${\rm det}\hskip 2.84526ptG_{i}=-1$, $d_{2}$ and $d_{3}$ would differ from the previous choices only by an overall minus sign.

Given this basic set up, we first discuss the general $\mu\tau$ interchange symmetry in the framework of residual $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$. We then proceed to the discussion of TBM mixing by setting the solar mixing angle $\theta_{12}=\sin^{-1}(1/\sqrt{3})$ in the $\mu\tau$ interchange scheme. A neutrino Majorana mass matrix

$\displaystyle M_{\nu}^{\mu\tau}=\begin{pmatrix}a&b&-b\\ b&c&d\\ -b&d&c\end{pmatrix}$

(2.3)

invariant under the $\mu\tau$ interchange symmetry is diagonalized as

$\displaystyle(U_{\nu}^{\mu\tau})^{T}M_{\nu}^{\mu\tau}U_{\nu}^{\mu\tau}=M_{d}^{\mu\tau},$

(2.4)

where

$\displaystyle U_{\nu}^{\mu\tau}=\begin{pmatrix}\sqrt{1-\lambda^{2}}&\lambda&0\\ -\frac{\lambda}{\sqrt{2}}&\frac{1}{\sqrt{2}}\sqrt{1-\lambda^{2}}&\frac{1}{\sqrt{2}}\\ \frac{\lambda}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\sqrt{1-\lambda^{2}}&\frac{1}{\sqrt{2}}\end{pmatrix}.$

(2.5)

The parameter $\lambda$ is related to the solar mixing angle as $\lambda=\sin\theta_{12}$. Here we choose the appropriate minus signs in $M_{\nu}^{\mu\tau}$ to be in conformity with the PDG convention[28]. Now from (2.2), $G_{2}$ and $G_{3}$ corresponding to $d_{2}$ and $d_{3}$ can be calculated as

$\displaystyle G_{2}^{\lambda}=\begin{pmatrix}2\lambda^{2}-1&\lambda\sqrt{2(1-\lambda^{2})}&-\lambda\sqrt{2(1-\lambda^{2})}\\ \lambda\sqrt{2(1-\lambda^{2})}&-\lambda^{2}&-(1-\lambda^{2})\\ -\lambda\sqrt{2(1-\lambda^{2})}&-(1-\lambda^{2})&-\lambda^{2}\end{pmatrix},G_{3}=\begin{pmatrix}-1&0&0\\ 0&0&1\\ 0&1&0\end{pmatrix}.$

(2.6)

The relation $d_{i}=d_{j}d_{k}$ makes the construction of $G_{1}$ simple: $G_{1}^{\lambda}=G_{2}^{\lambda}G_{3}$. Thus $G_{1}^{\lambda}$ will be of form

$\displaystyle G_{1}^{\lambda}=\begin{pmatrix}1-2\lambda^{2}&-\lambda\sqrt{2(1-\lambda^{2})}&\lambda\sqrt{2(1-\lambda^{2})}\\ -\lambda\sqrt{2(1-\lambda^{2})}&-(1-\lambda^{2})&-\lambda^{2}\\ \lambda\sqrt{2(1-\lambda^{2})}&-\lambda^{2}&-(1-\lambda^{2})\end{pmatrix}.$

(2.7)

Since $G_{3}$ is basically the $\mu\tau$ interchange symmetry in the flavour basis, we therefore rename the residual Klein symmetry for this $\mu\tau$ interchange case as $\mathbb{Z}_{2}^{\lambda}\times\mathbb{Z}_{2}^{\mu\tau}$. We now implement the $\mathbb{Z}_{2}^{\lambda}\times\mathbb{Z}_{2}^{\mu\tau}$ on the Dirac mass matrix $m_{D}$ and the Majorana mass matrix $M_{R}$ of (1.1) as

	$\displaystyle(G_{2}^{\lambda})^{T}m_{D}^{0}G_{2}^{\lambda}=m_{D}^{0},G_{3}^{T}m_{D}^{0}G_{3}=m_{D}^{0},$
	$\displaystyle(G_{2}^{\lambda})^{T}M_{R}^{0}G_{2}^{\lambda}=M_{R}^{0},G_{3}^{T}M_{R}^{0}G_{3}=M_{R}^{0}.$		(2.8)

Equations in (2.8) automatically imply the $G_{1}^{\lambda}$ invariance of $m_{D}$ and $M_{R}$ on account of the relation $G_{1}^{\lambda}=G_{2}^{\lambda}G_{3}$. Now one can work out the constraint equations that arise due the invariance relations in (2.8). A most general $3\times 3$ mass matrix

$\displaystyle M^{G}=\begin{pmatrix}A&B&C\\ D&E&F\\ G&H&I\end{pmatrix}$

(2.9)

that is invariant under $G^{\lambda}_{1,2,3}$ (cf. Eq. 2.8) would lead to the following constraint equations: for $G_{1}^{\lambda}$ invariance,

	$\displaystyle B+C$	$\displaystyle=$	$\displaystyle\lambda(\sqrt{2(1-\lambda^{2})})^{-1}(-H+F+E-I),$
	$\displaystyle D+G$	$\displaystyle=$	$\displaystyle\lambda(\sqrt{2(1-\lambda^{2})})^{-1}(H-F+E-I),$
	$\displaystyle(D-G+B-C)$	$\displaystyle=$	$\displaystyle\Big{\{}\sqrt{2(1-\lambda^{2})}(2\lambda^{2}-1)\Big{\}}^{-1}[4\lambda(1-\lambda^{2})A+2\lambda(1-\lambda^{2})(H+F-E-I)],$
	$\displaystyle B-C$	$\displaystyle=$	$\displaystyle D-G$		(2.10)

for $G_{2}^{\lambda}$ invariance,

	$\displaystyle B+C$	$\displaystyle=$	$\displaystyle-(2\lambda^{-1})\sqrt{2(1-\lambda^{2})}(-H+F+E-I),$
	$\displaystyle D+G$	$\displaystyle=$	$\displaystyle(2\lambda^{-1})\sqrt{2(1-\lambda^{2})}(-H+F-E+I),$
	$\displaystyle(D-G+B-C)$	$\displaystyle=$	$\displaystyle\Big{\{}\sqrt{2(1-\lambda^{2})}(2\lambda^{2}-1)\Big{\}}^{-1}[4\lambda(1-\lambda^{2})A+2\lambda(1-\lambda^{2})(H+F-E-I)],$
	$\displaystyle B-C$	$\displaystyle=$	$\displaystyle D-G$		(2.11)

and for $G_{3}^{\lambda}$ invariance,

$\displaystyle C=-B,G=-D,E=I,F=H.$

(2.12)

Note that since $m_{D}^{0}$ is a general complex $3\times 3$ matrix, one can simply consider the constraint equations derived in (2.10), (2.11) and (2.12). However, since $M_{R}^{0}$ is Majorana type, one has to consider a complex symmetric structure for the matrix $M_{G}$ in (2.9) which would require the replacements $B=D$, $C=G$ and $F=H$ in the equations (2.10), (2.11) and (2.12). The invariance equations of (2.8) are the consequences of an assumed $\mathbb{Z}_{2}^{\lambda}\times\mathbb{Z}_{2}^{\mu\tau}$ symmetry on both the left ($\nu_{L}$) and the right chiral ($\nu_{R}$) fields. It is worthwhile to highlight another interesting aspect of this symmetry. The overall invariance of the effective $M_{\nu}$ that arises from the Type-I seesaw mechanism, can be realized by implementing the residual $\mathbb{Z}_{2}^{\lambda}\times\mathbb{Z}_{2}^{\mu\tau}$ on the left-chiral fields only. Since $M_{\nu}$ arises due to the seesaw relation in (1.2), the invariance condition on $m_{D}$ alone

$\displaystyle G_{i}^{T}m_{D}^{0}=-m_{D}^{0}$

(2.13)

implies

$\displaystyle G_{i}^{T}M_{\nu}G_{i}=M_{\nu}.$

(2.14)

For such an invariance, the determinant of $m_{D}^{0}$ would be vanishing and therefore one of the neutrinos will become massless[33]. Proceeding in the similar manner, as in the previous case, we derive the following constraint equations for partial $G^{\lambda}_{123}$ invariance (cf. Eq. 2.13). For $G_{1}^{\lambda}$ invariance we have

	$\displaystyle 2(1-\lambda^{2})A-\lambda\sqrt{2(1-\lambda^{2})}(D-G)$	$\displaystyle=$	$\displaystyle 0,$
	$\displaystyle 2(1-\lambda^{2})B-\lambda\sqrt{2(1-\lambda^{2})}(E-H)$	$\displaystyle=$	$\displaystyle 0,$
	$\displaystyle 2(1-\lambda^{2})C-\lambda\sqrt{2(1-\lambda^{2})}(F-I)$	$\displaystyle=$	$\displaystyle 0.$		(2.15)

Similarly for $G_{2}^{\lambda}$ invariance

	$\displaystyle 2\lambda^{2}A+\lambda\sqrt{2(1-\lambda^{2})}(D-G)$	$\displaystyle=$	$\displaystyle 0,$
	$\displaystyle 2\lambda^{2}B+\lambda\sqrt{2(1-\lambda^{2})}(E-H)$	$\displaystyle=$	$\displaystyle 0,$
	$\displaystyle 2\lambda^{2}C+\lambda\sqrt{2(1-\lambda^{2})}(F-I)$	$\displaystyle=$	$\displaystyle 0$		(2.16)

and for $G_{3}^{\lambda}$ invariance

$\displaystyle D=-G,\hskip 2.84526ptE=-H,\hskip 2.84526ptF=-I~$

(2.17)

could be obtained. Again the minus signs in the invariance equations are used to be consistent with the PDG convention. Let us now switch to the analysis on TBM mixing which is a trivial generalization of the above discussion with $\lambda=1/\sqrt{3}$. We write the $\mathbb{Z}_{2}$ generators for TBM mixing as

$\displaystyle G_{1}^{TBM}=\frac{1}{3}\begin{pmatrix}1&-2&2\\ -2&-2&-1\\ 2&-1&-2\end{pmatrix},G_{2}^{TBM}=\frac{1}{3}\begin{pmatrix}-1&2&-2\\ 2&-1&-2\\ -2&-2&-1\end{pmatrix},G_{3}^{\mu\tau}=\begin{pmatrix}-1&0&0\\ 0&0&1\\ 0&1&0\end{pmatrix}.$

(2.18)

Similarly, for $\lambda=1/\sqrt{3}$, the well known $U_{TBM}$ mixing simply comes out from (2.5) as

$\displaystyle U_{TBM}=\begin{pmatrix}\sqrt{\frac{2}{3}}&\sqrt{\frac{1}{3}}&0\\ -\sqrt{\frac{1}{6}}&\sqrt{\frac{1}{3}}&\sqrt{\frac{1}{2}}\\ \sqrt{\frac{1}{6}}&-\sqrt{\frac{1}{3}}&\sqrt{\frac{1}{2}}\end{pmatrix}.$

(2.19)

As mentioned in the introduction, since $G_{3}^{\mu\tau}$ leads to a vanishing value $\theta_{13}$, the full $\mathbb{Z}_{2}^{\lambda}\times\mathbb{Z}_{2}^{\mu\tau}$ can not be a phenomenologically accepted symmetry of the Lagrangian. To be more precise, the nondegenarate eigenvalue of $d_{3}$, i.e., $(d_{3})_{33}=+1$, fixes the third column of $U_{\nu}^{\mu\tau}$ to $(0,1/\sqrt{2},1/\sqrt{2})^{T}$ which implies a vanishing value of $\theta_{13}$ while a nonzero value of the latter has been confirmed by the experiments at $5.2\sigma$[35]. Thus to generate a nonzero $\theta_{13}$ we break the $\mathbb{Z}_{2}^{\mu\tau}$ ($G_{3}^{\mu\tau}$) with small breaking parameters keeping the other residual $\mathbb{Z}_{2}$s (either $G_{1}^{TBM}$ or $G_{2}^{TBM}$) intact. In the next section, depending upon the residual symmetries on the neutrino fields and their breaking pattern, we categorize our discussion into three categories.

The residual TBM-Klein symmetry is implemented in the basis where $M_{R}$ is diagonal. This further leads to degenerate heavy RH neutrinos. We then consider the most general perturbation matrix that breaks only the $\mathbb{Z}_{2}^{\mu\tau}$ in $M_{R}$. Since these breaking parameters are responsible for generation of nonzero $\theta_{13}$, the extent of quasidegeneracy between the right handed neutrinos (or in other words the smallness of the breaking parameters) is now dictated by the $3\sigma$ value of $\theta_{13}$. A systematic discussion of the breaking scheme is presented in the following subsections.