Modified TBM and role of a hidden $\mathbb{Z}_2$

TL;DR

This paper explores minimally perturbed Majorana neutrino mass matrices within a residual _2 d7 _2 symmetry framework, deriving parameter constraints and demonstrating baryogenesis via flavored leptogenesis with quasi-degenerate RH neutrinos.

Contribution

It introduces a novel approach combining residual _2 d7 _2 symmetry with minimal perturbations to study neutrino masses and baryogenesis.

Findings

Derived constraint relations among neutrino parameters

Demonstrated baryogenesis via flavored leptogenesis

Highlighted the role of a hidden _2 symmetry

Abstract

In a residual $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry approach, we investigate minimally perturbed Majorana neutrino mass matrices. Constraint relations among the low energy neutrino parameters are obtained. Baryogenesis is realized through flavored leptogenesis mechanism with quasi-degenerate right handed (RH) heavy neutrinos.

Equations

$$-{\mathcal{L}}_{mass}^{\nu ,N}={\bar{\nu }}_{L\alpha }(m_D{)}_{\alpha i}{N}_{Ri}+\frac{1}{2}{\overline{N^C}}_{iR}(M_R{)}_i{\delta }_{ij}{N}_{jR}+h.c.$$

$$-{\mathcal{L}}_{mass}^{\nu }=\frac{1}{2}\overline{{\nu }_{L\alpha }^C}(M_{\nu }{)}_{\alpha \beta }{\nu }_{L\beta }+h.c..$$

$$m_D^0=\left(\begin{array}{ccc}b-c-a& a& -a\\ a& b& c\\ -a& c& b\end{array}\right),M_R^0=\left(\begin{array}{ccc}y& 0& 0\\ 0& y& 0\\ 0& 0& y\end{array}\right),M_R^{G_1ϵ}=\left(\begin{array}{ccc}0& ϵ& {ϵ}^{\prime }\\ ϵ& {ϵ}_4& 0\\ {ϵ}^{\prime }& 0& {ϵ}_6\end{array}\right)$$

$${\sin }^2{\theta }_{12}=\frac{1}{3}(1-2{\tan }^2{\theta }_{13}).$$

$$m_D^0=\left(\begin{array}{ccc}a& \frac{1}{2}(b-c)& \frac{1}{2}(c-b)\\ a& b& c\\ -a& c& b\end{array}\right),M_R^0=\left(\begin{array}{ccc}x& 0& 0\\ 0& y& 0\\ 0& 0& y\end{array}\right),M_R^{ϵ}=\left(\begin{array}{ccc}0& 0& 0\\ 0& {ϵ}_4& 0\\ 0& 0& {ϵ}_6\end{array}\right).$$

$$\displaystyle\frac{1}{4\pi v^{2}\mathcal{H}_{ii}}\Big{[}\sum\limits_{j\neq i}g(x_{ij})\hskip 2.84526pt{\rm Im}\hskip 2.84526pt\mathcal{H}_{ij}(m_{D})^{\dagger}_{i\alpha}(m_{D})_{\alpha j}+\sum_{j\neq i}\frac{r{\rm Im}\hskip 2.84526pt{\mathcal{H}}_{ji}(m_{D})^{\dagger}_{i\alpha}({m_{D}})_{\alpha j}}{r^{2}+\frac{\mathcal{H}_{jj}}{16\pi^{2}v^{4}}}\Big{]}.$$

$$g(x_{ij})$$

Full text

1 Introduction

In the minimally extended Standard Model (SM) with singlet RH neutrino fields $N_{Ri}$, $\mathcal{O}$(eV) neutrino masses are generated through Type-I seesaw mechanism. Relevant Lagrangian for the latter can be written as

[TABLE]

with $N^{C}_{i}=C\bar{N}_{i}^{T}$. The effective light neutrino Majorana mass matrix $M_{\nu}$ which is then obtained by the standard seesaw formula $M_{\nu}=-m_{D}M_{R}^{-1}m_{D}^{T},$ can be put into a diagonal form as $U^{T}M_{\nu}U=M_{\nu}^{d}\equiv\rm diag\hskip 2.84526pt(m_{1},m_{2},m_{3})$ with $m_{i}$ assumed to be real. The effective low energy neutrino Majorana mass term that contains this $M_{\nu}$ comes out as

[TABLE]

Now in the basis where the charged lepton mass matrix $M_{\ell}$ is diagonal, $U$ follows the standard parametrization[1]. Let us now have look at the latest $3\sigma$ ranges [2] for the relevant neutrino parameters obtained from oscillation data. solar: $\Delta m_{21}^{2}\equiv$ $m_{2}^{2}-m_{1}^{2}$: $(7.02\textendash 8.09)\times 10^{-5}$ $\rm eV^{2}$, atmospheric: $|\Delta m_{31}^{2}|\equiv$ $|m_{3}^{2}-m_{1}^{2}|$: $(2.32\textendash 2.59)\times 10^{-3}$ $\rm eV^{2}$, $\theta_{12}$: $31.29^{o}\textendash 35.91^{o}$, $\theta_{23}$: $38.3^{o}\textendash 53.3^{o}$, $\theta_{13}$: $7.87^{o}\textendash 9.11^{o}$. Finally, thanks to the Planck for the observed upper bound on the sum of the light neutrino masses; $\Sigma_{i}m_{i}<$ 0.23 eV.

In Ref.[3] it is argued that any horizontal symmetry of neutrino Majorana mass matrix $M_{\nu}$ is a residual $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ flavour symmetry. The symmetry generators $G_{i}$ obey the relation $U^{\dagger}G_{i}U=d_{i}$ with i=2, 3, i.e. there are two independent $G_{i}$ and hence $d_{i}$. We can choose these two independent $d$ matrices as $d_{2}={\rm diag}\hskip 2.84526pt(-1,1,-1)$ and $d_{3}={\rm diag}\hskip 2.84526pt(-1,-1,1)$. Thus for a given $U$, one can calculate $G_{2}$ and $G_{3}$ corresponding to $d_{2}$ and $d_{3}$ respectively. In this work we focus particularly on the TBM mixing and calculate the corresponding $G_{i}$ matrices; $G_{1,2}^{TBM}$ and $G_{3}^{\mu\tau}$. It can be justified theoretically as well as phenomenologically that $G_{1}^{TBM}$ is the only symmetry which is viable one to exist as the unbroken $\mathbb{Z}_{2}$ generator in the Lagrangian. In the next section, we present an ephemeral discussion regarding the implementation of $G_{1}^{TBM}$ and $G_{3}^{\mu\tau}$ on the neutrino fields. For further insights related to the application of residual symmetry in the neutrino sector, the readers could have a quick look at Ref.[7].

2 Breaking of $\mathbb{Z}_{2}^{\mu\tau}$: perturbation to the TBM mass matrices

Depending upon the residual symmetries on the neutrino fields and the phenomenological viability of the textures of the mass matrices, we discuss two cases.

Case 1. At the leading order $G_{1}^{TBM}$ and $G_{3}^{\mu\tau}$ transform both the neutrino fields $\nu_{L}$ and $N_{R}$ as $\nu_{L}\rightarrow G_{i}\nu_{L}$ and $N_{R}\rightarrow G_{i}N_{R}$. Now we choose a perturbation matrix $M_{R}^{G_{1}\epsilon}$ which violates $\mu\tau$ interchange in $M_{R}$ but respect $G_{1}^{TBM}$. The leading order mass matrices and the perturbation matrix are of forms

[TABLE]

where $\epsilon=\frac{1}{4}(3\epsilon_{4}+\epsilon_{6})$ and $\epsilon^{\prime}=-\frac{1}{4}(3\epsilon_{6}+\epsilon_{4})$. Now the effective $M_{\nu}$ which is invariant under $G_{1}^{TBM}$ is written as $M_{\nu 1}^{G_{1}^{TBM}}=-m_{D}^{0}M_{R}^{-1}(m_{D}^{0})^{T}$ with $M_{R}=M_{R}^{0}+M_{R}^{G_{1}\epsilon}$. Since $G_{1}^{TBM}$ invariance of the effective $M_{\nu}$ always fixes the first column of the mixing matrix to $(\sqrt{\frac{2}{3}},-\sqrt{\frac{1}{6}},\sqrt{\frac{1}{6}})^{T}$ up to some phases, a direct comparison of the latter with the $U_{PMNS}$ matrix leads to a constraint relation between $\theta_{12}$ and $\theta_{13}$ as

[TABLE]

Case 2. In this case, at the leading order, all the neutrino fields obey $G_{3}^{\mu\tau}$. However $G_{1}^{TBM}$ of the over all $M_{\nu}$ is ensured only by the transformation $\nu_{L}\rightarrow G_{1}^{TBM}\nu_{L}$. Since the RH singlets are free from $G_{1}^{TBM}$, the perturbation matrix which is added with $M_{R}$ is now arbitrary. Now the most general Dirac mass matrix $m_{D}^{0}$, the Majorana mass matrix $M_{R}^{0}$ and the perturbation matrix are of the forms

[TABLE]

Again the effective $M_{\nu}$ is calculated as $M_{\nu 2}^{G_{1}^{TBM}}=-m_{D}^{0}M_{R}^{-1}(m_{D}^{0})^{T}$ with $M_{R}=M_{R}^{0}+M_{R}^{\epsilon}$. Besides reproducing the same relation as obtained in Eq. 4, another interesting point is realized that $m_{D}^{0}$ of (5) is of determinant zero due to the residual $G_{1}^{TBM}$ symmetry; thus the $M_{\nu 2}^{G_{1}^{TBM}}$ matrix has one zero eigenvalue. For the remnant $G_{1}^{TBM}$ symmetry, $m_{1}$ is of vanishing value.

One can also obtain correlation of $\delta$ with the mixing angles[8].

3 Flavored leptogenesis with quasi degenerate RH neutrinos

Lepton number, CP violating and out of equilibrium decays of RH neutrinos create a lepton asymmetry[4]. A general expression for the CP asymmetry parameter $\epsilon_{i}^{\alpha}$ for any RH mass spectrum is given by[5]

[TABLE]

In (6), $r=(1-x_{ij})$, $\mathcal{H}\equiv{m_{D}}^{\dagger}m_{D}$, $x_{ij}=M_{j}/M_{i}$ and $g(x_{ij})$ is given by

[TABLE]

The term proportional to $f(x_{ij})$ comes from the one loop vertex contribution while the remaining are from self energy diagram. Note that in the limit where the RH neutrinos are exactly degenerate, i.e., $x_{ij}=1$, the self energy contribution vanishes and thus a nonzero value of CP asymmetry parameter $\varepsilon^{\alpha}_{i}$ is produced only through the vertex contribution. In our model RH neutrinos are quasi degenerate and thereby enhances the CP asymmetry parameter significantly through self energy contributions.

Another important issue is that the flavor effect[6] to the produced lepton asymmetry. In [8] we address this issue in detail, theoretically as well as numerically. See Fig.2 for a typical variation of $Y_{B}$ with $z=M_{1}/T$.