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 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.

Global solution to the cubic Dirac equation in two space dimensions

“…An important relation between E D (s, ψ) and E + (s, ψ) was found in [7] (see Proposition 2.3), which reads…”

“…Finally, let us illustrate the energy estimates (see [7], Proposition 2.4), which shall play a crucial role in our proof.…”

“…Let us recall the energy estimates for the Dirac field on hyperboloids H s , which was first introduced in [7] (see Section 2.3). Given a complex-valued function ψ(t, x) : R 1+2 → C 2 on H s , we define the following energy functional…”

“…Next, let us prove the unified pointwise decay with respect to m ∈ [0, 1]. Using the semi-hyperboloidal frame, we can rewrite the Dirac equation as (see also [7])…”

Global stability of Minkowski spacetime for a spin-1/2 field

“…We also want to commute the vector fields with the Dirac equation γ µ D µ ψ = 0. Like [1] and [11], we modify the boost and rotation vector fields so that they commute with the flat Dirac operator.…”