Fully Constrained Majorana Neutrino Mass Matrices using $\Sigma (72\times 3)$

Abstract

In 2002, two neutrino mixing ansatze having trimaximally mixed middle (${\nu }_2$) columns, namely tri-chi-maximal mixing ($\text{T}\chi \text{M}$) and tri-phi-maximal mixing ($\text{T}\varphi \text{M}$), were proposed. It was recently shown that $\text{T}\chi \text{M}$ with $\chi =\pm \frac{\pi }{16}$ as well as $\text{T}\varphi \text{M}$ with $\varphi =\pm \frac{\pi }{16}$ leads to the solution, ${\sin }^2{\theta }_{13}=\frac{2}{3}{\sin }^2\frac{\pi }{16}$, consistent with the latest measurements of the reactor mixing angle, ${\theta }_{13}$. To obtain $\text{T}\chi {\text{M}}_{(\chi =\pm \frac{\pi }{16})}$ and $\text{T}\varphi {\text{M}}_{(\varphi =\pm \frac{\pi }{16})}$, we utilised the type I see-saw framework with fully constrained Majorana neutrino mass matrices. These mass matrices also resulted in a relation among the neutrino masses, $m_1:m_2:m_3=\frac{\left(2+\sqrt{2}\right)}{1+\sqrt{2(2+\sqrt{2})}}:1:\frac{\left(2+\sqrt{2}\right)}{-1+\sqrt{2(2+\sqrt{2})}}$. In this paper we construct a flavour model based on the discrete group $\Sigma (72\times 3)$ and obtain the aforementioned results. A Majorana neutrino mass matrix (a symmetric $3\times 3$ matrix with 6 complex degrees of freedom) is conveniently mapped into a flavon field transforming as the complex 6 dimensional representation of $\Sigma (72\times 3)$. Specific vacuum alignments of the flavons are used to arrive at the desired mass matrices.

Comments: 28 pages, 1 figure