  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.
