  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.
