The framework of the auxiliary group: two birds with one stone

Flavon models in the literature assume constraints on the components of the vacuum expectation values (vevs) of flavons, and typically, these constraints are not fully determined by the residual symmetry group of the set of vevs. This poses a problem because the general potential of the flavons cannot have a minimum that leads to such constraints unless additional mechanisms involving supersymmetry, extra dimensions etc., are invoked. In this paper, we show that the framework of the auxiliary group naturally results in vevs satisfying the required constraints, and using it, we construct a type-1 seesaw model with two right-handed neutrinos, which predicts the ratio of the light neutrino masses $m_2/m_3=(\sqrt{2}-1)/(\sqrt{2}+1)$ and $\text{TM}_1$ mixing with $\sin\theta_{13}=\frac{1}{\sqrt{3}}\sin\frac{\pi}{12}$ and $\sin\delta_\text{CP}=-1$. Our framework posits auxiliary group transformations which act on the flavons but not on the fermions. We construct the general renormalizable potential without invoking additional mechanisms and show that it has a minimum that leads to the required constraints.