A loophole in the proofs of asymptotic freedom and quantum triviality

Abstract: In 1973, Coleman and Gross proved that in four dimensions, only non-abelian gauge theories can have asymptotic freedom. More recently, Aizenman and Duminil-Copin proved that four dimensional scalar field theories are quantum trivial in the continuum. Both of these proofs have a loophole, and it is the same loophole in both proofs: The proofs assume that the scalar self-coupling in the UV is positive definite. While this is a perfectly reasonable and classically very intuitive assumption, it is an assumption nevertheless. In this work, I show that the assumption of coupling positivity is violated in a concrete quantum field theory, the O(N) model, in the large N limit. Surprisingly, despite the classically nonsensical unbounded potential, the negative coupling has no pathological consequence for propagators, the free energy or cross sections. This suggests that interacting scalar field theories with asymptotic freedom in four dimensions are possible, despite long-held opinions to the contrary.