微中子震盪實驗已經被量測,從實驗結果得知微中子是有質量的。質量的特徵函數跟味的特徵函數有所不同,中間利用微中子混合矩陣連接。最有名的混合矩陣為TBM 混合矩陣,但是他預測 θ13為零。以現今的實驗數據告訴我們 θ13有95%不為零。所以我們利用加入無效應的微中子去改變 θ13,並且與實驗數據比較。

The phenomenon of neutrino oscillation has been established. It implies
that neutrinos having masses, so the mass eigenstate is different with the
flavor eigenstate. They has mixing matrix described by a 3 × 3 unitary matrix,
PMNS, like CKM matrix in quark sector. The most famous pattern is
Tri-Bimaximal mixing matrix which predicts a vanishing θ13, but we have
experimental evidence for a non vanishing value of the smallest θ13. TriBimaximal
mixing matrix needs modifications. We study discrete flavour
symmetries model(A4 symmetry), higher symmetry model and sterile neutrinos
to explain small angle. First, suppose there has A4 symmetry or higher
symmetry in lepton sector and neutrino sector and neutrinos have Majorana
masses(or only have Dirac neutrinos). We will get Tri-Bimaximal mixing
in A4 model. Second, Adding sterile neutrinos modify the mixing matrix,
therefore we could get small angle(θ13). Then, transforming with new mixing
matrix, we found ranges of neutrino masses,mixing angle and so on. We
compare them to the experiment data

1 Introduction
2 Neutrino
2.1 Neutrino oscillations
2.2 Neutrino mass matrix
3 A4 Symmetry 12
3.1 The Alternating Group A4
3.2 A flavor model with A4
3.2.1 Tri-Bimaximal Lepton Mixing
3.2.2 A4-Model
3.3 A4 × U(1) × Z2 × Z2 × Z2 symmetry model
4 The Tri-Bimaximal neutrino mixing matrix could be modi-
fied by adding sterile neutrinos
4.1 Diagonalization of mass matrix
4.2 Model building with the sterile neutrinos
4.3 The mass matrix and mixing
4.4 Data comparing with experiment
5 Conclusion

[1] MH Ahn, E Aliu, S Andringa, S Aoki, Y Aoyama, J Argyriades,
K Asakura, R Ashie, F Berghaus, HG Berns, et al. Measurement
of neutrino oscillation by the k2k experiment. Physical Review D,
74(7):072003, 2006.
[2] Guido Altarelli and Ferruccio Feruglio. Tri-bimaximal neutrino mixing
from discrete symmetry in extra dimensions. Nuclear Physics B,
720(1):64–88, 2005.
[3] Stefan Antusch, Carla Biggio, Enrique Fern´andez-Mart´ınez, M Belen
Gavela, and Jacobo L´opez-Pav´on. Unitarity of the leptonic mixing matrix.
Journal of High Energy Physics, 2006(10):084, 2006.
[4] Samoil M Bilenky and B Pontecorvo. Lepton mixing and neutrino oscillations.
Physics Reports, 41(4):225–261, 1978.
[5] DV Forero, M Tortola, and JWF Valle. Neutrino oscillations refitted.
Physical Review D, 90(9):093006, 2014.
[6] M Concepci´on Gonz´alez-Garci´a. Theory of neutrino masses and mixing.
Nuclear Physics B-Proceedings Supplements, 117:186–203, 2003.
[7] Paul F Harrison, Don H Perkins, and WG Scott. Tri-bimaximal mixing
and the neutrino oscillation data. Physics Letters B, 530(1):167–173,
2002.
[8] M Je˙zabek and Y Sumino. Neutrino mixing and see-saw mechanism.
Physics Letters B, 440(3):327–331, 1998.
[9] Boris Kayser. Neutrino mass, mixing, and flavor change. In Neutrino
Mass, pages 1–24. Springer, 2003.
[10] LING-FONG LI and TA-PEI CHENG. Gauge theory of elementary
particle physics. first edition edition, 2000.
[11] JL Raaf, Super-Kamiokande Collaboration, et al. Solar and atmospheric
neutrinos in super-kamiokande. In Journal of Physics: Conference Series,
volume 136, page 022013. IOP Publishing, 2008.
[12] Zhen-hua Zhao. Minimal modifications to the tri-bimaximal neutrino
mixing. Journal of High Energy Physics, 2014(11):1–20, 2014.