Paul Romatschke: A loophole in the proofs of quantum triviality and asymptotic freedom

Back in 1973, Sidney Coleman and Nobel Prize winner David Gross proved that only non-abelian gauge theories can have asymptotic freedom. More recently, Michael Aizenman and Hugo Duminil-Copin (Field Medalist 2022) proved that scalar field theories are quantum trivial in four dimensions. Both of these important proofs have the same loophole: they assume that the coupling constant in the UV is positive definite. While a reasonable physics assumption, the results obtained in non-Hermitian (PT-symmetric) quantum mechanics indicate that certain field theories with negative coupling exist and fulfill the basic requirements of unitarity and stability. I will show an explicit calculation on how a particular field theory (the O(N) model) in 3+1 dimensions exploits this loophole, leading to a candidate for an asymptotically free interacting scalar field theory, and a potential alternative to the Standard Model Higgs sector.