Geometry of the Ground State of Higgs Fields in Next-to-MSSM

Decomposing the Higgs potential ${\mathcal{V}}_{higgs}$ of next - to - Minimal Supersymmetric Standard Model (n-MSSM) \ as the sum of three contributions like ${\mathcal{V}}_{ch}+{\mathcal{V}}_{kah}+{\mathcal{V}}_{expl}$ and assuming the two following things: $\left(a\right)$ ${\mathcal{V}}_{higgs}$ dominated by ${\mathcal{V}}_{ch}$ coming from the chiral sector of supersymmetry: ${\mathcal{V}}_{ch}={\mid \nu \mid }^2\mathcal{U}$ with $\mid \nu \mid$ large and $\frac{$\mathrm{r}$}{\left \vert $\nu$\right \vert }<<1$;and$\left( b\right) $ replacing the chiral down Higgs superfield doublet ${\left({\mathit{H}}_d\right)}^i$ of n-MSSM by a chiral anti-doublet ${\left({\mathbf{\Phi }}_d\right)}_i$; we derive the explicit geometry of the Higgs fields in the ground state $\left \vert \Sigma _{higgs}\right \rangle $ found to be given by two intersecting conifolds. We show as well that the property $\tan {\beta }_{susy}=1$ living at singularity $r=0$ is a supersymmetric signal; and deviation away reads in terms of the Kahler parameter $r$ and the ${\vartheta }_{{}_W}$- Weinberg angle as $\tan \beta \simeq1+\frac{$\mathrm{r}$}{2\left \vert $\nu$\right \vert }\sin^{2}\vartheta {{W}}$. Other related issues are also studied.