# Global evolution of the U(1) Higgs Boson: nonlinear stability and uniform energy bounds

## TL;DR

This paper proves the nonlinear stability of the U(1) Higgs boson model by establishing global energy bounds and decay results for coupled wave-Klein-Gordon and Dirac equations using hyperboloidal foliation methods.

## Contribution

It introduces a new energy functional for the Dirac equation and demonstrates uniform decay and energy bounds for the Higgs and gauge fields in a nonlinear stability framework.

## Key findings

- Proved nonlinear stability of the U(1) Higgs model.
- Established uniform energy bounds for Higgs and gauge fields.
- Derived decay results for the Dirac equation independent of mass.

## Abstract

Relying on the hyperboloidal foliation method, we establish the nonlinear stability of the ground state of the $U(1)$ standard model of electroweak interactions. This amounts to establishing a global-in-time theory for the initial value problem for a nonlinear wave-Klein-Gordon system that couples (Dirac, scalar, gauge) massive equations together. In particular, we investigate here the Dirac equation and consider a new energy functional for this field defined with respect to the hyperboloidal foliation of Minkowski spacetime. We provide a novel decay result for the Dirac equation which is uniform in the mass coefficient, and thus allows for the Dirac mass coefficient to be arbitrarily small. Furthermore we obtain energy bounds for the Higgs fields and gauge bosons that are uniform with respect to the hyperboloidal time variable.