Neutrino mixing parameters and masses from ∆(96) ⋊ HCP in the tri-direct CP approach

A bstract We present a comprehensive model independent analysis of all breaking patterns resulting from ∆(96) ⋊ H CP in the tri-direct CP approach of the minimal seesaw model with two right-handed neutrinos. The three generations of left-handed lepton doublets are assumed to transform as the irreducible triplet 3 0 of ∆(96), and the two right-handed neutrinos are assigned to singlets. In the case that both flavon fields ϕ atm and ϕ sol transform as triplet $$ {\overline{\textbf{3}}}_{\textbf{0}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mn>3</mml:mn> <mml:mo>¯</mml:mo> </mml:mover> <mml:mn>0</mml:mn> </mml:msub> </mml:math> , only one phenomenologically viable lepton mixing pattern is obtained for normal ordering neutrino masses. The lepton mixing matrix is predicted to be the TM1 pattern, with neutrino masses, mixing angles, and CP violation phases depending on only three real input parameters. When ϕ sol is assigned to the $$ {\overline{\textbf{3}}}_{\textbf{1}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mn>3</mml:mn> <mml:mo>¯</mml:mo> </mml:mover> <mml:mn>1</mml:mn> </mml:msub> </mml:math> representation, an additional real parameter x must be included. Then we find 42 (12) independent phenomenologically interesting mixing patterns for normal (inverted) ordering neutrino masses, and the corresponding predictions for lepton mixing parameters and neutrino masses are obtained. Furthermore, we perform a detailed numerical analysis for five (one) example breaking patterns with some benchmark values of x for normal (inverted) ordering. For the five normal examples, the absolute values of the first columns of the PMNS matrix are fixed to be $$ {\left(\sqrt{\frac{2}{3}},\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}}\right)}^T $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mfenced> <mml:msqrt> <mml:mfrac> <mml:mn>2</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> </mml:msqrt> <mml:mfrac> <mml:mn>1</mml:mn> <mml:msqrt> <mml:mn>6</mml:mn> </mml:msqrt> </mml:mfrac> <mml:mfrac> <mml:mn>1</mml:mn> <mml:msqrt> <mml:mn>6</mml:mn> </mml:msqrt> </mml:mfrac> </mml:mfenced> <mml:mi>T</mml:mi> </mml:msup> </mml:math> , $$ \frac{1}{5}{\left(\sqrt{17},2,2\right)}^T $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>5</mml:mn> </mml:mfrac> <mml:msup> <mml:mfenced> <mml:msqrt> <mml:mn>17</mml:mn> </mml:msqrt> <mml:mn>2</mml:mn> <mml:mn>2</mml:mn> </mml:mfenced> <mml:mi>T</mml:mi> </mml:msup> </mml:math> , $$ \frac{1}{\sqrt{38}}{\left(5,2,3\right)}^T $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:msqrt> <mml:mn>38</mml:mn> </mml:msqrt> </mml:mfrac> <mml:msup> <mml:mfenced> <mml:mn>5</mml:mn> <mml:mn>2</mml:mn> <mml:mn>3</mml:mn> </mml:mfenced> <mml:mi>T</mml:mi> </mml:msup> </mml:math> , $$ \frac{1}{\sqrt{57}}{\left(\sqrt{37},\sqrt{10},\sqrt{10}\right)}^T $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:msqrt> <mml:mn>57</mml:mn> </mml:msqrt> </mml:mfrac> <mml:msup> <mml:mfenced> <mml:msqrt> <mml:mn>37</mml:mn> </mml:msqrt> <mml:msqrt> <mml:mn>10</mml:mn> </mml:msqrt> <mml:msqrt> <mml:mn>10</mml:mn> </mml:msqrt> </mml:mfenced> <mml:mi>T</mml:mi> </mml:msup> </mml:math> and $$ \frac{1}{3}{\left(\sqrt{6},1,\sqrt{2}\right)}^T $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> <mml:msup> <mml:mfenced> <mml:msqrt> <mml:mn>6</mml:mn> </mml:msqrt> <mml:mn>1</mml:mn> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> </mml:mfenced> <mml:mi>T</mml:mi> </mml:msup> </mml:math> , respectively. For the inverted example, the absolute value of the third column of the PMNS matrix is $$ \frac{1}{2\sqrt{11}}{\left(1,5,3\sqrt{2}\right)}^T $$ <mml:math xmlns:m