Fully constrained Majorana neutrino mass matrices using $$\varvec{\varSigma (72\times 3)}$$ Σ(72×3)

Abstract In 2002, two neutrino mixing ansatze having trimaximally mixed middle ($$\nu _2$$ ν2 ) columns, namely tri-chi-maximal mixing ($$\text {T}\chi \text {M}$$ TχM ) and tri-phi-maximal mixing ($$\text {T}\phi \text {M}$$ TϕM ), were proposed. In 2012, it was shown that $$\text {T}\chi \text {M}$$ TχM with $$\chi =\pm \,\frac{\pi }{16}$$ χ=±π16 as well as $$\text {T}\phi \text {M}$$ TϕM with $$\phi = \pm \,\frac{\pi }{16}$$ ϕ=±π16 leads to the solution, $$\sin ^2 \theta _{13} = \frac{2}{3} \sin ^2 \frac{\pi }{16}$$ sin2θ13=23sin2π16 , consistent with the latest measurements of the reactor mixing angle, $$\theta _{13}$$ θ13 . To obtain $$\text {T}\chi \text {M}_{(\chi =\pm \,\frac{\pi }{16})}$$ TχM(χ=±π16) and $$\text {T}\phi \text {M}_{(\phi =\pm \,\frac{\pi }{16})}$$ TϕM(ϕ=±π16) , 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, $$m_1:m_2:m_3=\frac{\left( 2+\sqrt{2}\right) }{1+\sqrt{2(2+\sqrt{2})}}:1:\frac{\left( 2+\sqrt{2}\right) }{-1+\sqrt{2(2+\sqrt{2})}}$$ m1:m2:m3=2+21+2(2+2):1:2+2-1+2(2+2) . In this paper we construct a flavour model based on the discrete group $$\varSigma (72\times 3)$$ Σ(72×3) and obtain the aforementioned results. A Majorana neutrino mass matrix (a symmetric $$3\times 3$$ 3×3 matrix with six complex degrees of freedom) is conveniently mapped into a flavon field transforming as the complex six-dimensional representation of $$\varSigma (72\times 3)$$ Σ(72×3) . Specific vacuum alignments of the flavons are used to arrive at the desired mass matrices.