Dirac CP violation phase in the neutrino sector with A4 flavour symmetry

Author affiliations


  • Phi Quang Van Institute of Physics, Vietnam Academy of Science and Technology, 10 Dao Tan, Hanoi 11108, Vietnam
  • Nguyen Anh Ky Institute of Physics, Vietnam Academy of Science and Technology, 10 Dao Tan, Hanoi 11108, Vietnam https://orcid.org/0000-0003-0471-197X
  • Tien Manh Tran Institute of Physics, Vietnam Academy of Science and Technology, 10 Dao Tan, Ba Dinh, Hanoi 11108, Vietnam https://orcid.org/0009-0002-6463-4902




CP viol, Jarlskog invariant,, neutrino mass, perturbation


CP violation is one of the problems of the physics beyond the Standard Model. It can happen in both the quark and the lepton sectors. In the present paper, following the work arXiv:1602.07437 [hep-ph], this problem is re-considered in the lepton sector (neutrino subsector) within an extended Standard Model with an $A_4$ flavour discrete symmetry with a new and more convenient parametrization. As a result, a perturbative mixing matrix is derived. Then, the Dirac CP violation phase \(\delta_{CP}\equiv \delta\) and the Jarlskog invariant \(J_{CP}\equiv J\) are analytically obtained from theoretically derived equations leading to the solutions \(\delta= \pm {1\over 2}\pi\) . Between the two solutions, the solution \(\delta=-{1\over 2}\pi\) (i.e., \({3\over 2}\pi\)) is more preferable as it is more consistent with the experimental data for the inverted ordering of the neutrino masses for the gobal fit [PDG] or the normal ordering [T2K, NO\(\nu\)A]). A relation between $\delta$ and $J$ is also given in terms of new parameters. The maximum value of Jarlskog invariant \(|J^{max}|\) is found in the range $0.0237 < |J^{max}| < 0.034$, covering the 2022-2023 global fit values [PDG]: \(|J_{PDG}^{max}|= 0.0336\pm 0.0006 ~(\pm 0.0019)\) at \(1\sigma ~(3\sigma\)). Other values of J can be determined by the ralation \(J (\delta)\) and approximated by Fig. 2. between two solutions.


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How to Cite

Q. V. Phi, A. K. Nguyen, and T. M. Tran, “Dirac CP violation phase in the neutrino sector with A4 flavour symmetry”, Comm. Phys., vol. 34, no. 2, p. 125, Jun. 2024.



Received 28-02-2024
Accepted 19-05-2024
Published 11-06-2024