# Geometry of Flat Directions in Scale-Invariant Potentials

## TL;DR

This paper introduces a novel geometric method to identify flat directions in scale-invariant scalar potentials by analyzing the determinant and hyperdeterminant conditions of coupling matrices, simplifying complex multi-field studies.

## Contribution

It proposes a new geometric framework based on determinants and hyperdeterminants to find flat directions in scale-invariant potentials, applicable to many fields.

## Key findings

- Flat directions occur when the determinant of the coupling matrix vanishes.
- The hyperdeterminant condition generalizes the approach to arbitrary quartic potentials.
- Application examples include scalar extensions of the Standard Model.

## Abstract

We observe that biquadratic potentials admit non-trivial flat directions when the determinant of the quartic coupling matrix of the scalar fields vanishes. This consideration suggests a new approach to the problem of finding flat directions in scale-invariant theories, noticeably simplifying the study of scalar potentials involving many fields. The method generalizes to arbitrary quartic potentials by requiring that the hyperdeterminant of the tensor of scalar couplings be zero. We demonstrate our approach with detailed examples pertaining to common scalar extensions of the Standard Model.