Naturalness in see-saw mechanism and Bogoliubov transformation

Abstract

We present an alternative perspective on the see-saw mechanism for the neutrino mass, according to which the small neutrino mass is given as a difference of two large masses. This view emerges when an analogue of the Bogoliubov transformation is used to describe Majorana neutrinos in the Lagrangian of the see-saw mechanism, which is analogous to the BCS theory. The Bogoliubov transformation clarifies the natural appearance of Majorana fermions when C is strongly violated by the right-handed neutrino mass term with good CP in the single flavor model. Analyzing typical models with $m_{R}$= $10^{4}$ to $10^{15}$ GeV, it is shown that a hitherto unrecognized fine tuning of the order $m_{\nu}/m_{R}=10^{-15}$ to $10^{-26}$ is required to make the commonly perceived see-saw mechanism work in a natural setting, namely, when none of dimensionless coupling constants are very small.