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: $\mathcal{V}_{ch}=\left \vert \mathrm{\nu }\right \vert ^{2}\mathcal{U}$ with $\left \vert \mathrm{\nu }\right \vert $ large and $\frac{\mathrm{r}}{\left \vert \mathrm{\nu }\right \vert }<<1$; and $\left( b\right) $ replacing the chiral down Higgs superfield doublet $\left( \boldsymbol{H}_{d}\right) ^{i}$ of n-MSSM by a chiral anti-doublet $\left( \boldsymbol{\Phi}_{d}\right) _{i}$; 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 $\tan \beta \simeq1+\frac{\mathrm{r}}{2\left \vert \mathrm{\nu }\right \vert }\sin^{2}\vartheta _{_{W}}$. Other related issues are also studied.