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

In this paper, we search for conditions under which the mass matrix of light neutrinos m_ν is not a result of large cancellations for the type-I seesaw mechanism with four-zero texture. For the Yukawa matrix of neutrinos Y_ν 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 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 m_ν 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, Y_ν must have a cascade hierarchy. For M_R, the 12 submatrix has a similar hierarchy to Y_ν and m_ν. However, the 23 submatrix has a waterfall hierarchy without some fine-tuning. Therefore, it is likely that Y_ν and M_R 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.