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 study a new energy functional defined with respect to the hyperboloidal foliation of Minkowski spacetime. We provide a 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 establish energy bounds for the Higgs fields and gauge bosons that are uniform with respect to the hyperboloidal time variable.
Highlights
We view this model as a stepping stone toward the full nonabelian Glashow–Weinberg– Salam theory (GSW), known as the electroweak standard model
We study here a class of nonlinear wave equations that involves the first-order Dirac equation coupled to second-order wave or Klein–Gordon equations
We provide here a new application of the hyperboloidal foliation method introduced for such coupled systems by LeFloch and Ma [19], which has been successfully used to establish global-in-time existence results for nonlinear systems of coupled wave and Klein–Gordon equations
Summary
We provide here a new application of the hyperboloidal foliation method introduced for such coupled systems by LeFloch and Ma [19], which has been successfully used to establish global-in-time existence results for nonlinear systems of coupled wave and Klein–Gordon equations. Ψ behaves more like a wave component when mg 1, but since the mass mg may be very small but nonzero, we cannot apply techniques for wave equations We find it possible to overcome these difficulties by analyzing the first-order Dirac equation, which admits the positive energy functional (1.13), and this energy plays a key role in the whole analysis.
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