# Dirac CP phase in the neutrino mixing matrix and the Froggatt-Nielsen mechanism with ${\rm \bf det} \bf [M_\nu]=0$

## TL;DR

This paper predicts the Dirac CP phase in neutrino mixing within a Froggatt-Nielsen framework with a zero determinant condition, linking it to mixing angles and neutrinoless double beta decay.

## Contribution

It introduces a model imposing a zero determinant condition on the neutrino mass matrix, drastically constraining the Dirac CP phase and correlating it with mixing angles.

## Key findings

- Predicted Dirac CP phase in the range of ±(0.4-2.9) radians.
- Correlation between δ_CP and sin²θ₂₃, with δ_CP approaching ±π/2 for larger sin²θ₂₃.
- Effective mass m_ee for neutrinoless double beta decay estimated between 3.3 and 4.0 meV.

## Abstract

We discuss the Dirac CP violating phase $\delta_{CP}$ in the Froggatt-Nielsen model for a neutrino mass matrix $M_\nu$ imposing a condition ${\rm det} [M_\nu]=0$. This additional condition restricts the CP violating phase $\delta_{CP}$ drastically. We find that the phase $\delta_{CP}$ is predicted in the region of $\pm (0.4- 2.9)$ radian, which is consistent with the recent T2K and NO$\nu$A data. There is a remarkable correlation between $\delta_{CP}$ and $\sin^2\theta_{23}$. The phase $\delta_{CP}$ converges on $\sim \pm \pi/2$ if $\sin^2\theta_{23}$ is larger than $0.5$. Thus, accurate measurements of $\sin^2\theta_{23}$ are important for a test of our model. The effective mass $m_{ee}$ for the neutrinoless double beta decay is predicted in the rage of $3.3-4.0$ meV.