Conditions of naturalness and fine-tunings for type-I seesaw mechanism with four-zero texture

TL;DR

This paper investigates naturalness conditions for the neutrino mass matrices in the type-I seesaw mechanism with four-zero textures, revealing alignment conditions, hierarchical structures, and implications for CP symmetry.

Contribution

It introduces alignment conditions that prevent large cancellations in neutrino mass matrices and explores their implications for flavor structures and CP symmetry in the seesaw mechanism.

Findings

Alignment conditions relate Yukawa and Majorana matrices.

Hierarchical structures are constrained by these conditions.

A universal generalized CP symmetry is implied.

Abstract

In this paper, we search for conditions that the mass matrix of light neutrinos $m_{\nu}$ is not a result of large cancellations for the type-I seesaw mechanism with four-zero texture. For the Yukawa matrix of neutrinos $Y_{\nu}$ and heavy Majorana mass matrix $M_{R}$, these conditions are written as $(Y_{\nu})_{i2} \propto (m_{\nu})_{i2} \, \Rightarrow \, (Y_{\nu})_{i2} \propto (M_{R})_{i2}$. We call them {\it alignment} conditions because they align the certain rows or columns of the three neutrino mass matrices. If these conditions do not hold, the large mixing in $m_{\nu}$ is a result of fine-tuning due to the cancellation of several terms. Then they are required from a viewpoint of naturalness. They give an explanation of the seesaw-invariance of four-zero texture, and place rough restrictions on flavor structures of neutrinos. Under these conditions, $Y_{\nu}$ must have a cascade hierarchy. For $M_{R}$, the 12 submatrix has a similar hierarchy as $Y_{\nu}$ and $m_{\nu}$. However, the 23 submatrix has a waterfall hierarchy without some fine-tuning. Therefore, it is likely that $Y_{\nu}$ and $M_{R}$ have qualitatively different flavor structures. Furthermore, since the conditions restrict CP phases of the matrix elements, they imply existence of a universal generalized CP symmetry in the neutrino sector.