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Fully Constrained Majorana Neutrino Mass Matrices using $Σ(72\times3)$

In 2002, two neutrino mixing ansatze having trimaximally mixed middle ($ν_2$) columns, namely tri-chi-maximal mixing ($\text{T}χ\text{M}$) and tri-phi-maximal mixing ($\text{T}ϕ\text{M}$), were proposed. It was recently shown that $\text{T}χ\text{M}$ with $χ=\pm \fracπ{16}$ as well as $\text{T}ϕ\text{M}$ with $ϕ= \pm \fracπ{16}$ leads to the solution, $\sin^2 θ_{13} = \frac{2}{3} \sin^2 \fracπ{16}$, consistent with the latest measurements of the reactor mixing angle, $θ_{13}$. To obtain $\text{T}χ\text{M}_{(χ=\pm \fracπ{16})}$ and $\text{T}ϕ\text{M}_{(ϕ=\pm \fracπ{16})}$, 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 $Σ(72\times3)$ and obtain the aforementioned results. A Majorana neutrino mass matrix (a symmetric $3\times3$ matrix with 6 complex degrees of freedom) is conveniently mapped into a flavon field transforming as the complex 6 dimensional representation of $Σ(72\times3)$. Specific vacuum alignments of the flavons are used to arrive at the desired mass matrices.