General solutions for TM$_{1,2}$ mixing in the minimal type-I seesaw mechanism and phenomenological constraints to Yukawa matrix of neutrinos

TL;DR

This paper derives a general solution for TM$_{1,2}$ mixing in the minimal type-I seesaw mechanism, linking symmetry constraints to phenomenological parameters and reducing the free parameters in the neutrino Yukawa matrix.

Contribution

It provides a concise, symmetry-based parametrization of the Yukawa matrix in the minimal seesaw model, connecting theoretical constraints with observable neutrino parameters.

Findings

Only one free complex parameter remains in the Yukawa matrix after constraints.

Symmetry conditions restrict parameter dependence on right-handed neutrino masses.

Physical parameters for TM$_{1}$ with normal hierarchy depend only on certain Yukawa combinations.

Abstract

In this paper, we concisely represent the general solution for the TM$_{1,2}$ mixing in the type-I minimal seesaw mechanism and show phenomenological constraints of the Yukawa matrix of neutrinos $Y$. First, conditions of $Z_{2}$ symmetry associated with TM$_{1,2}$ restrict two complex parameters. In addition, three complex parameters are constrained from five phenomenological parameters (two neutrino masses $m_{i}$, the mixing angle $\theta_{13}$, the Dirac phase $\delta$, and the Majorana phase $\alpha$) and the overall phase. As a result, there is only one free complex parameter for $Y$, except for the right-handed neutrino masses $M_{1,2}$. For the TM$_{1}$ mixing with normal hierarchy, the symmetry condition is imposed only on $Y$, and the physical parameters depend on $M_{R}$. In other situations, the dimensionless parameters $(\theta_{13}, \delta$ and $\alpha)$ do not depend on $M_{R}$ but only on certain combinations of the components of $Y$.