Algorithmic Boundedness-From-Below Conditions for Generic Scalar Potentials

TL;DR

This paper introduces a novel algorithmic approach using multivariate algebra and tensor spectral theory to determine whether scalar potentials are bounded from below, applicable to complex models with extended scalar sectors.

Contribution

It presents a new computational method for establishing BFB conditions in polynomial scalar potentials, leveraging advanced algebraic and tensor spectral techniques.

Findings

Developed a Mathematica implementation for BFB analysis

Illustrated the approach with elementary examples

Potential to derive analytical BFB conditions for complex models

Abstract

Checking that a scalar potential is bounded from below (BFB) is an ubiquitous and notoriously difficult task in many models with extended scalar sectors. Exact analytic BFB conditions are known only in simple cases. In this work, we present a novel approach to algorithmically establish the BFB conditions for any polynomial scalar potential. The method relies on elements of multivariate algebra, in particular, on resultants and on the spectral theory of tensors, which is being developed by the mathematical community. We give first a pedagogical introduction to this approach, illustrate it with elementary examples, and then present the working Mathematica implementation publicly available at GitHub. Due to the rapidly increasing complexity of the problem, we have not yet produced ready-to-use analytical BFB conditions for new multi-scalar cases. But we are confident that the present implementation can be dramatically improved and may eventually lead to such results.