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

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.