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

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.

Comments: 16 pages, will be published in PTEP