  Standard Model with a classical conformal invariance holds the promise to
give a better understanding of the hierarchy problem and could pave the way for
beyond the standard model physics. So, we give here a mathematical treatment of
a massless quartic scalar field theory with a strong self-coupling both
classically and for quantum field theory. We use a set of classical solutions
recently found and show that there exists an infinite set of infrared trivial
scalar theories with a mass gap. Free particles have superimposed a harmonic
oscillator set of states. The classical solution is displayed through a current
expansion and the next-to-leading order quantum correction is provided.
Application to the Standard Model would entail the existence of higher excited
states of the Higgs particle and reduced decay rates to WW and ZZ that could be
already measured.
