# Non-Minimal Approximation for the Type-I Seesaw Mechanism

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. A Model with Three Right-Handed Neutrinos

#### 2.1. Standard Approximation

#### 2.2. Non-Minimal Approximation

#### 2.3. Parametrization of the Mixing Matrix $\Theta $

#### 2.3.1. Standard Approximation

#### 2.3.2. Non-Minimal Approximation

#### 2.4. Interactions of the Majorana Fields with SM Particles

## 3. Numerical Examples

- BP
_{min}: - ${m}_{0}=0$, ${M}_{1}$ = 50 keV, M = 100 GeV,
- BP
_{max}: - ${m}_{0}={10}^{-5}$ eV, ${M}_{1}$ = 1 keV, M = 1 GeV.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

SM | Standard model of particle physics |

GUT | Grand unified theory |

$\nu $MSM | Neutrino minimal extension of the SM |

DM | Dark matter |

HNL | Heavy neutral lepton |

BAU | Baryon asymmetry in the Universe |

PMNS | Pontecorvo–Maki–Nakagawa–Sakata |

LH (RH) | Left-handed (right-handed) |

NO (IO) | Normal ordering (inversed ordering) |

## Appendix A. Symmetric Matrices m_{ν}, M_{N}, M_{M} and Relations for M_{D}θ^{T} and θ^{†}M_{D}

## Appendix B. A Simple Representation for the Matrix Ω_{nm}

## Appendix C. The Effective Masses of Active Neutrinos in Standard and Non-Minimal Approximations

## References

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**Figure 1.**Mixing parameter estimations for NO: (

**a**) the dependence of ${U}^{2}$ as a function of masses of heavy neutrinos ${M}_{1}$ and ${M}_{2}={M}_{3}=M$ at ${m}_{0}={10}^{-5}$ eV, ${\Omega}_{\mathrm{m}}={\Omega}^{{}^{\prime}}$; (

**b**) the relative difference in ${U}^{2}$ obtained in minimal and non-minimal approximations as a function of ${M}_{2}={M}_{3}=M$ at BP1, ${m}_{0}=0$, ${M}_{1}$ = 1 keV in units of ${10}^{-12}$ (black dotted line); BP2, ${m}_{0}={10}^{-5}$ eV, ${M}_{1}$ = 1 keV in units of ${10}^{-9}$ (red dashed line); BP3, ${m}_{0}=0$, ${M}_{1}$ = 50 keV in units of ${10}^{-12}$ (green solid line); and BP4, ${m}_{0}={10}^{-5}$ eV, ${M}_{1}$ = 50 keV in units of ${10}^{-11}$ (blue dot-dashed line).

**Table 1.**Numerical estimations of diagonal elements ${\left({m}_{\nu}\right)}_{ee}$, ${\left({m}_{\nu}\right)}_{\mu \mu}$ and ${\left({m}_{\nu}\right)}_{\tau \tau}$ in units of ${10}^{-2}$ eV for explicit forms ${\Omega}_{\mathrm{m}}$: ${\Omega}^{{}^{\prime}},{\Omega}^{{}^{\u2033}},{\Omega}^{{}^{\u2034}}$, see (52). Here, ${m}_{0}$ and ${M}_{1},{M}_{2},{M}_{3}$ are defined by (49) and (50).

Ordering | $\mathit{l}=\mathit{e}$ | $\mathit{l}=\mathit{\mu}$ | $\mathit{l}=\mathit{\tau}$ |
---|---|---|---|

NO | $0.4$ | $2.7$ | $2.6$ |

IO | $4.9$ | $1.9$ | $2.6$ |

Ordering | ${\mathbf{\Omega}}^{{}^{\prime}}$ | ${\mathbf{\Omega}}^{{}^{\u2033}}$ | ${\mathbf{\Omega}}^{{}^{\u2034}}$ |
---|---|---|---|

NO | $\phantom{\rule{4pt}{0ex}}{10}^{-12}-{10}^{-8}$ | ${10}^{-7}-{10}^{-5}$ | ${10}^{-12}-{10}^{-8}$ |

IO | ${10}^{-6}-{10}^{-5}$ | ${10}^{-6}-{10}^{-5}$ | ${10}^{-12}-{10}^{-8}$ |

**Table 3.**The relative difference $|{U}_{\mathrm{nm}}^{2}-{U}_{\mathrm{m}}^{2}|/{U}_{\mathrm{nm}}^{2}$ for NO, IO and different ${\Omega}_{\mathrm{m}}$ (${\Omega}^{{}^{\prime}},{\Omega}^{{}^{\u2033}},{\Omega}^{{}^{\u2034}}$) (52) at BP${}_{\mathrm{min}}$ (BP${}_{\mathrm{max}}$).

Ordering | ${\mathbf{\Omega}}^{{}^{\prime}}$ | ${\mathbf{\Omega}}^{{}^{\u2033}}$ | ${\mathbf{\Omega}}^{{}^{\u2034}}$ |
---|---|---|---|

NO | $\phantom{\rule{4pt}{0ex}}{10}^{-13}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{4.pt}{0ex}}\left({10}^{-9}\right)$ | ${10}^{6}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{4.pt}{0ex}}\left({10}^{3}\right)$ | ${10}^{-13}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{4.pt}{0ex}}\left({10}^{-9}\right)$ |

IO | ${10}^{-7}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{4.pt}{0ex}}\left({10}^{-5}\right)$ | $0.1\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{4.pt}{0ex}}\left(0.1\right)$ | $1\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{4.pt}{0ex}}\left(1\right)$ |

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Dubinin, M.; Fedotova, E.
Non-Minimal Approximation for the Type-I Seesaw Mechanism. *Symmetry* **2023**, *15*, 679.
https://doi.org/10.3390/sym15030679

**AMA Style**

Dubinin M, Fedotova E.
Non-Minimal Approximation for the Type-I Seesaw Mechanism. *Symmetry*. 2023; 15(3):679.
https://doi.org/10.3390/sym15030679

**Chicago/Turabian Style**

Dubinin, Mikhail, and Elena Fedotova.
2023. "Non-Minimal Approximation for the Type-I Seesaw Mechanism" *Symmetry* 15, no. 3: 679.
https://doi.org/10.3390/sym15030679