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Traditionally, scalar $\phi^4$ theory in four dimensions is thought to bequantum trivial in the continuum. This tradition is apparently well groundedboth in physics arguments and mathematical proofs. Digging into the proofs onefinds that they do not actually cover all physically meaningful situations, inparticular the case of multi-component fields and non-polynomial action. Inthis work, I study multi-component scalar field theories in four dimensions inthe continuum and show that they do evade the apparently foregone conclusion oftriviality. Instead, one finds a non-trivial interacting theory that has twophases, bound states and non-trivial scattering amplitudes in the limit of manycomponents. This has potentially broad implications, both for the foundationsof quantum field theory as well as for the experimentally accessible Higgssector of the Standard Model.

What if $\phi^4$ theory in 4 dimensions is non-trivial in the continuum?