The Vacuum Moduli Space of the Minimal Supersymmetric Standard Model

Abstract

A starting point in the study of the minimal supersymmetric Standard Model (MSSM) is the vacuum moduli space, which is a highly complicated algebraic variety: it is the image of an affine variety $X\subset {\mathbb{C}}^{49}$ under a symplectic quotient map $\ddot{I}•$ to ${\mathbb{C}}^{973}$. Previous work computed the vacuum moduli space of the electroweak sector; geometrically this corresponds to studying a restriction of $\ddot{I}•$: ${\mathbb{C}}^{13}\stackrel{\ddot{I}{•}^{\text{res}}}{⟶}{\mathbb{C}}^{22}$. We analyze the geometry of the full vacuum moduli space for superpotentials $W_{\mathrm{minimal}}$ (without neutrinos) and $W_{\mathrm{MSSM}}$ (with neutrinos) in ${\mathbb{C}}^{973}$. In both cases, we prove that $X$ consists of three irreducible components $X_1$, $X_2$, and $X_3$, and determine the images $M_i$ of the $X_i$ under $\ddot{I}•$. For $W_{\mathrm{minimal}}$ we show they have, respectively, dimensions $1$, $15$, and $29$, and prove that each of the $M_i$ is a rational variety, while for $W_{\mathrm{MSSM}}$ we show that $M_3$ is the only component. Restricting the $M_i$ to the electroweak sector, we recover known results. We describe the components of the vacuum moduli space geometrically in terms of incidence varieties to a product of Segre varieties.

41 pages, 3 figures