Vacuum Geometry of the Standard Model

TL;DR

This paper uses algebraic geometry and computational techniques to explicitly determine the vacuum structure of the minimal supersymmetric Standard Model, revealing its complex geometric components.

Contribution

It applies Gr"obner bases and symmetry exploitation to compute the vacuum moduli space of the MSSM, a novel algebraic geometric analysis of this fundamental theory.

Findings

The vacuum moduli space has three irreducible components with dimensions 1, 15, and 29.

Each component is a rational variety defined by explicit algebraic equations.

The geometric structure encodes solutions to F-terms and D-terms in gauge invariant operators.

Abstract

Vacuum structure of a quantum field theory is a crucial property. In theories with extended symmetries, such as supersymmetric gauge theories, the vacuum is typically a continuous manifold, called the vacuum moduli space, parametrized by the expectation values of scalar fields. Starting from the R-parity preserving superpotential at renormalizable order, we use Gr\"obner bases to determine the explicit structure, as an algebraic variety, of the vacuum geometry of the minimal supersymmetric extension of the Standard Model. Gr\"obner bases have doubly exponential computational complexity (for this case, $7^{2^{1023}}$ operations); we exploit symmetry and multigrading to render the computation tractable. This geometry has three irreducible components of complex dimensions $1$, $15$, and $29$, each being a so-called rational variety. The defining equations of the components express the solutions to F-terms and D-terms in terms of the gauge invariant operators and are interpreted in terms of classical geometric constructions.