  Traditionally, scalar $\phi^4$ theory in four dimensions is thought to be
quantum trivial in the continuum. This tradition is apparently well grounded
both in physics arguments and mathematical proofs. Digging into the proofs one
finds that they do not actually cover all physically meaningful situations, in
particular the case of multi-component fields and non-polynomial action. In
this work, I study multi-component scalar field theories in four dimensions in
the continuum and show that they do evade the apparently foregone conclusion of
triviality. Instead, one finds a non-trivial interacting theory that has two
phases, bound states and non-trivial scattering amplitudes in the limit of many
components. This has potentially broad implications, both for the foundations
of quantum field theory as well as for the experimentally accessible Higgs
sector of the Standard Model.
