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In 1973, Coleman and Gross proved that in four dimensions, only non-abeliangauge theories can have asymptotic freedom. More recently, Aizenman andDuminil-Copin proved that four dimensional scalar field theories are quantumtrivial in the continuum. Both of these proofs have a loophole, and it is thesame loophole in both proofs: The proofs assume that the scalar self-couplingin the UV is positive definite. While this is a perfectly reasonable andclassically very intuitive assumption, it is an assumption nevertheless. Inthis work, I show that the assumption of coupling positivity is violated in aconcrete quantum field theory, the O(N) model, in the large N limit.Surprisingly, despite the classically nonsensical unbounded potential, thenegative coupling has no pathological consequence for propagators, the freeenergy or cross sections. This suggests that interacting scalar field theorieswith asymptotic freedom in four dimensions are possible, despite long-heldopinions to the contrary.

A loophole in the proofs of asymptotic freedom and quantum triviality