# Hidden relations in three generation seesaw model with Dirac mass matrix of four-zero texture

## TL;DR

This paper investigates CP violation in a Type-I seesaw model with four-zero textures, revealing hidden relations among Majorana mass matrix elements that constrain CP phases and enhance understanding of neutrino properties.

## Contribution

It introduces a comprehensive method to analyze four-zero textures in the Type-I seesaw model, uncovering hidden relations that link Majorana matrix elements to CP violating phases.

## Key findings

- Possible ranges of CP phases are numerically demonstrated.
- Strong correlations among CP phases are identified.
- Hidden relations among Majorana matrix elements are classified and explained.

## Abstract

We present predictions for CP violating phases of the Type-I seesaw model with four-zero textures on the Dirac mass matrix. For the four-zero textures, the effective low energy Majorana mass matrix is parametrized with seven parameters. They are three mass-dimensional parameters, two angles and two CP violating sources. The number of these parameters is less than that of the general description of the Majorana mass matrix with three neutrino masses, three mixing angles and three CP violating phases. In particular, only two independent CP violating sources give rise to three CP violating phases. The efficient and comprehensive method is proposed in this paper to investigate four-zero textures in Type-I seesaw. We numerically show the possible range of CP violating phases in the plane of a Dirac CP violating phase and one of Majorana phases. Some cases show the strong correlations among two phases. These correlations can be explained by hidden relations among the elements of Majorana matrix with the four-zero textures. The hidden relations are classified according to the position of one vanishing off-diagonal element of Majorana mass matrix. The Majorana mass matrix all of whose elements are non-vanishing also produces other hidden relations particularly in the case of four-zero textures. By applying the hidden relations, we describe the concrete correlations among CP violating phases.