# Three Flavor Quasi-Dirac Neutrino Mixing, Oscillations and Neutrinoless Double Beta Decay

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## Abstract

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## 1. Introduction

## 2. Theory

## 3. Consequences of Our Specific QD Scenario for Processes Measuring Neutrino Masses

## 4. The Survival Probabilities of Electron Antineutrino

- We see that ${P}_{{\overline{\nu}}_{e}\to {\overline{\nu}}_{e}}(\u03f5\ne 0)-{P}_{{\overline{\nu}}_{e}\to {\overline{\nu}}_{e}}(\u03f5=0)$ is directly proportional to $L/E$ ratio multiplied with $\u03f5$. Thus, as we go to higher $L/E$, the effect of $\u03f5$ becomes larger.
- Since ${\theta}_{13}$ is very small when compared to ${\theta}_{12}$, the term ${m}_{3}^{2}{sin}^{4}{\theta}_{13}$ in second part of Equation (38) becomes negligible as compared to two other terms containing ${m}_{1}$ and ${m}_{2}$, respectively. As a result, the second part of Equation (38) in the case of NO (${m}_{1}<{m}_{2}<{m}_{3}$) is always smaller than that in IO scenario (${m}_{3}<{m}_{1}<{m}_{2}$). Therefore, we expect the modification in survival probability due to non-zero $\u03f5$ to be smaller when the mass pattern is NO than that for IO.

## 5. Constraints on Majorana Component of Neutrino Masses

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The survival probabilities of ${\overline{\nu}}_{e}$ as the functions of energy E for 1.5 km (upper panel), 53 km (middle panel), and 180 km (lower panel) baselines. The plots in left and right panels are with NO and IO as mass orderings, respectively. The probabilities are shown for three cases: $\left(i\right)$$\u03f5=0$, $3\nu $ mixing case, $\left(ii\right)$$\u03f5={10}^{-4}$ eV, and $\left(iii\right)$$\u03f5=2\times {10}^{-4}$ eV, with black, green, and red lines, respectively. In cases (ii) and (iii), the lightest Dirac mass is 0.01 eV. The cyan bands represent ${\overline{\nu}}_{e}$ survival probabilities in three-flavor neutrino oscillation framework with $3\sigma $ allowed range of oscillation parameters. The benchmark values of oscillation parameters along and their $3\sigma $ allowed ranges, as used in this study, are given in Table 1.

**Figure 2.**The top (bottom) panels show the allowed ranges of lightest Dirac mass and $\u03f5$ that are obtained using 1.5 km baseline and 4 MeV (8 MeV) energy. The left and right panels are obtained with NO and IO, respectively. In all of the panels, pink (cyan) and red (blue) lines correspond to minimum and maximum ${\overline{\nu}}_{e}$ survival probabilities, respectively, allowed in three-flavor neutrino oscillation picture with $1\sigma $ ($3\sigma $) uncertainty of oscillation parameters. The green line corresponds to ${\overline{\nu}}_{e}$ survival probability in $3\nu $ framework with the benchmark value of oscillation parameters. For details, see text. The benchmark values of oscillation parameters along with their $1\sigma $ and $3\sigma $ allowed ranges, used in this study, are given in Table 1.

Parameters | Best Fit Values | $1\mathit{\sigma}$ Range | $3\mathit{\sigma}$ Range |
---|---|---|---|

${\theta}_{12}$ (${}^{\xb0}$) | 34 | [33.1, 34.6] | [31, 37] |

${\theta}_{13}$ (${}^{\xb0}$) | 8.5 | [8.48, 8.74] | [40, 53] |

$|\Delta {m}_{31}^{2}/{10}^{-3}|$ (eV${}^{2}$) | 2.5 | [2.49, 2.55] | [2.4, 2.6] |

$\Delta {m}_{21}^{2}/{10}^{-5}$ (eV${}^{2}$) | 7.5 | [7.2,7.6] | [6.7, 8.0] |

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**MDPI and ACS Style**

Khatun, A.; Smetana, A.; Šimkovic, F.
Three Flavor Quasi-Dirac Neutrino Mixing, Oscillations and Neutrinoless Double Beta Decay. *Symmetry* **2020**, *12*, 1310.
https://doi.org/10.3390/sym12081310

**AMA Style**

Khatun A, Smetana A, Šimkovic F.
Three Flavor Quasi-Dirac Neutrino Mixing, Oscillations and Neutrinoless Double Beta Decay. *Symmetry*. 2020; 12(8):1310.
https://doi.org/10.3390/sym12081310

**Chicago/Turabian Style**

Khatun, Amina, Adam Smetana, and Fedor Šimkovic.
2020. "Three Flavor Quasi-Dirac Neutrino Mixing, Oscillations and Neutrinoless Double Beta Decay" *Symmetry* 12, no. 8: 1310.
https://doi.org/10.3390/sym12081310