Vacuum Stability Conditions and Potential Minima for a Matrix Representation in Lightcone Orbit Space

TL;DR

This paper introduces a Minkowski space-based formalism to analyze vacuum stability and potential minima for scalar fields in complex matrix representations, applicable to models like left-right symmetric theories.

Contribution

It develops a novel method using orbit space and Minkowski structure to determine stability conditions for complex matrix scalar potentials, extending previous approaches.

Findings

Derived vacuum stability conditions for left-right symmetric potential

Applied the formalism to bidoublet and Higgs doublet models

Provided a systematic approach for potential minimization in matrix representations

Abstract

The orbit space for a scalar field in a complex square matrix representation obtains a Minkowski space structure from the Cauchy-Schwarz inequality. It can be used to find vacuum stability conditions and minima of the scalar potential. The method is suitable for fields such as a bidoublet, an $SU(2)$ triplet or $SU(3)$ octet. We use the formalism to find the vacuum stability conditions for the left-right symmetric potential of a bidoublet and left and right Higgs doublets.