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

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 ($\text{T}\phi\text{M}$), were proposed. In 2012, it was shown that $\text{T}\chi\text{M}$ with $\chi=\pm \frac{\pi}{16}$ as well as $\text{T}\phi\text{M}$ with $\phi = \pm \frac{\pi}{16}$ 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 $\text{T}\phi\text{M}_{(\phi=\pm \frac{\pi}{16})}$, the type~I see-saw framework with fully constrained Majorana neutrino mass matrices was utilised. These mass matrices also resulted in the neutrino mass ratios, $m_1:m_2:m_3=\frac{\left(2+\sqrt{2}\right)}{1+\sqrt{2(2+\sqrt{2})}}:1:\frac{\left(2+\sqrt{2}\right)}{-1+\sqrt{2(2+\sqrt{2})}}$. 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.