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

Abstract

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