# A Method to Explore Flavor Symmetries of the 3HDM and Their Implications on Lepton Masses and Mixing

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Flavor Symmetry of the 3HDM

## 3. Solving the Invariance Equations

## 4. Method to Find Relations among the Lepton Mass and Mixing Parameters

Algorithm 1: Obtain Dirac mass matrices from group symmetry of the 3HDM. | ||||||

1 | Let S be a set of finite groups of interest; | |||||

2 | for every G in Sdo | |||||

3 | if3 divides the order of Gthen | |||||

4 | obtain character table of G; | |||||

5 | obtain E, the set of characters for three-dimensional irreducible representations of G; | |||||

6 | forevery 4-tuple of characters $(A,B,C,H)$ in ${E}^{4}$do | |||||

7 | ifone of $A,B,H$ is faithful
and ${n}_{1}$:=innerProduct($A\otimes {B}^{*},{H}^{*}$) = 1 then | |||||

8 | Obtain unitary representation matrices for the 3D irreps of $A,B,H$; | |||||

9 | forevery generator of Gdo | |||||

10 | setup Kronecker product equation; | |||||

11 | end | |||||

12 | Solve set of Kronecker product equations for ${h}^{l}$-matrices | |||||

13 | Calculate charged lepton mass matrix ${M}^{l}$ | |||||

14 | end | |||||

15 | ifone of $A,C,H$ is faithful and ${n}_{2}$:= innerProduct($A\otimes {C}^{*},H$) = 1 then | |||||

16 | Obtain unitary representation matrices for the 3D irreps of $A,C,H$ | |||||

17 | forevery generator of Gdo | |||||

18 | setup Kronecker product equation; | |||||

19 | end | |||||

20 | Solve set of Kronecker product equations for ${h}^{\nu}$-matrices | |||||

21 | Calculate Dirac neutrino mass matrix ${M}^{\nu}$; | |||||

22 | end | |||||

23 | if${n}_{1}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1$and${n}_{2}=1$and one of$A,B,H$is faithful | |||||

24 | and one of$A,C,H$is faithful | |||||

25 | then | |||||

26 | Calculate ${U}_{PMNS}$ from ${M}^{l}$ and ${M}^{\nu}$; | |||||

27 | end | |||||

28 | end | |||||

29 | end | |||||

30 | end |

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Harari, H. Three Generations of Quarks and Leptons. In Proceedings of the 5th International Conference on Meson Spectroscopy, Boston, MA, USA, 29–30 April 1977; p. 0170. [Google Scholar]
- Ibe, M.; Kusenko, A.; Yanagida, T.T. Why three generations? Phys. Lett. B
**2016**, 758, 365–369. [Google Scholar] [CrossRef] - King, S.F.; Merle, A.; Morisi, S.; Shimizu, Y.; Tanimoto, M. Neutrino Mass and Mixing: From Theory to Experiment. New J. Phys.
**2014**, 16, 045018. [Google Scholar] [CrossRef] - Rong, S.J. Lepton Mixing Patterns from PSL
_{2}(7) with a Generalized CP Symmetry. Adv. High Energy Phys.**2020**, 2020, 6120803. [Google Scholar] [CrossRef] - Joshipura, A.S.; Patel, K.M. Residual Z
_{2}symmetries and leptonic mixing patterns from finite discrete subgroups of U(3). JHEP**2017**, 1, 134. [Google Scholar] [CrossRef] - Qian, X.; Vogel, P. Neutrino Mass Hierarchy. Prog. Part. Nucl. Phys.
**2015**, 83, 1–30. [Google Scholar] [CrossRef] - Frampton, P.H.; Kephart, T.W. Simple nonAbelian finite flavor groups and fermion masses. Int. J. Mod. Phys.
**1995**, A10, 4689–4704. [Google Scholar] [CrossRef] - Aker, M.; Beglarian, A.; Behrens, J.; Berlev, A.; Besserer, U.; Bieringer, B.; Block, F.; Bobien, S.; Boettcher, M.; Bornschein, B.; et al. Direct neutrino-mass measurement with sub-electronvolt sensitivity. Nat. Phys.
**2022**, 18, 160–166. [Google Scholar] [CrossRef] - Abe, K.; Akutsu, R.; Ali, A.; Alt, C.; Andreopoulos, C.; Anthony, L.; Antonova, M.; Aoki, S.; Ariga, A.; Asada, Y.; et al. Constraint on the matter–antimatter symmetry-violating phase in neutrino oscillations. Nature
**2020**, 580, 339–344. [Google Scholar] [CrossRef] - Bak, G.; Choi, J.H.; Jang, H.I.; Jang, J.S.; Jeon, S.H.; Joo, K.K.; Ju, K.; Jung, D.E.; Kim, J.G.; Kim, J.H.; et al. Measurement of Reactor Antineutrino Oscillation Amplitude and Frequency at RENO. Phys. Rev. Lett.
**2018**, 121, 201801. [Google Scholar] [CrossRef] - Feruglio, F.; Romanino, A. Lepton flavor symmetries. Rev. Mod. Phys.
**2021**, 93, 015007. [Google Scholar] [CrossRef] - De Adelhart Toorop, R.; Feruglio, F.; Hagedorn, C. Finite Modular Groups and Lepton Mixing. Nucl. Phys. B
**2012**, 858, 437–467. [Google Scholar] [CrossRef] - Holthausen, M.; Lim, K.S.; Lindner, M. Lepton Mixing Patterns from a Scan of Finite Discrete Groups. Phys. Lett. B
**2013**, 721, 61–67. [Google Scholar] [CrossRef] - Parattu, K.M.; Wingerter, A. Tribimaximal Mixing From Small Groups. Phys. Rev.
**2011**, D84, 013011. [Google Scholar] [CrossRef] - Lam, C.S. Group Theory and Dynamics of Neutrino Mixing. Phys. Rev. D
**2011**, 83, 113002. [Google Scholar] [CrossRef] - Lam, C.S. Finite Symmetry of Leptonic Mass Matrices. Phys. Rev. D
**2013**, 87, 013001. [Google Scholar] [CrossRef] - Branco, G.; Ferreira, P.; Lavoura, L.; Rebelo, M.; Sher, M.; Silva, J.P. Theory and phenomenology of two-Higgs-doublet models. Phys. Rep.
**2012**, 516, 1–102. [Google Scholar] [CrossRef] - Chaber, P.; Dziewit, B.; Holeczek, J.; Richter, M.; Zralek, M.; Zajac, S. Lepton masses and mixing in a two-Higgs-doublet model. Phys. Rev. D
**2018**, 98, 055007. [Google Scholar] [CrossRef] - Dziewit, B.; Holeczek, J.; Zajac, S.; Zralek, M. Family Symmetries and Multi Higgs Doublet Models. Symmetry
**2020**, 12, 156. [Google Scholar] [CrossRef] - Grossman, Y. Phenomenology of models with more than two Higgs doublets. Nucl. Phys. B
**1994**, 426, 355–384. [Google Scholar] [CrossRef] - Keus, V.; King, S.F.; Moretti, S. Three-Higgs-doublet models: Symmetries, potentials and Higgs boson masses. JHEP
**2014**, 1, 052. [Google Scholar] [CrossRef] [Green Version] - Ivanov, I.P.; Vdovin, E. Classification of finite reparametrization symmetry groups in the three-Higgs-doublet model. Eur. Phys. J. C
**2013**, 73. [Google Scholar] [CrossRef] - Weinberg, S. Baryon and Lepton Nonconserving Processes. Phys. Rev. Lett.
**1979**, 43, 1566–1570. [Google Scholar] [CrossRef] - Ludl, P.O. On the finite subgroups of U(3) of order smaller than 512. J. Phys.
**2010**, A43, 395204, Erratum in J. Phys. A**2011**, 44, 139501. [Google Scholar] [CrossRef] [Green Version] - The GAP Group. GAP—Groups, Algorithms, and Programming, Version 4.11.1. 2021. Available online: https://www.gap-system.org (accessed on 5 August 2022).
- Dabbaghian, V. Repsn, A GAP4 Package for Constructing Representations of Finite Groups, Version 3.0.2. Refereed GAP Package. 2011. Available online: http://www.sfu.ca/~vdabbagh/gap/repsn.html (accessed on 5 August 2022).
- Wolfram Inc. Mathematica, Version 12.3.1; Wolfram Inc.: Champaign, IL, USA, 2021; Available online: https://www.wolfram.com/mathematica (accessed on 5 August 2022).

**Figure 1.**${U}_{PMNS}$ mixing angles for ${S}_{4}$-symmetry; $|{v}_{i}/{v}_{1}|\in [0,1.2]$. The shaded areas indicate $1\sigma $ intervals of the quantities. The red points deviate less than $1\sigma $ from the experimental values of sin${}^{2}{\theta}_{13}$ and ${\delta}_{CP}$.

1 | 6 | 8 | 3 | 6 | |

${\chi}^{1}$ | 1 | 1 | 1 | 1 | 1 |

${\chi}^{2}$ | 1 | −1 | 1 | 1 | −1 |

${\chi}^{3}$ | 2 | 0 | −1 | 2 | 0 |

${\chi}^{4}$ | 3 | −1 | 0 | −1 | 1 |

${\chi}^{5}$ | 3 | 1 | 0 | −1 | −1 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Dziewit, B.; Zrałek, M.; Vergeest, J.; Chaber, P.
A Method to Explore Flavor Symmetries of the 3HDM and Their Implications on Lepton Masses and Mixing. *Symmetry* **2022**, *14*, 1854.
https://doi.org/10.3390/sym14091854

**AMA Style**

Dziewit B, Zrałek M, Vergeest J, Chaber P.
A Method to Explore Flavor Symmetries of the 3HDM and Their Implications on Lepton Masses and Mixing. *Symmetry*. 2022; 14(9):1854.
https://doi.org/10.3390/sym14091854

**Chicago/Turabian Style**

Dziewit, Bartosz, Marek Zrałek, Joris Vergeest, and Piotr Chaber.
2022. "A Method to Explore Flavor Symmetries of the 3HDM and Their Implications on Lepton Masses and Mixing" *Symmetry* 14, no. 9: 1854.
https://doi.org/10.3390/sym14091854