An all-order discontinuity at the electroweak phase transition

We define a non-local gauge-invariant Greenʹs function which can distinguish between the symmetric (confinement) and broken (Higgs) phases of the hot SU(2)×U(1) electroweak theory to all orders in the perturbative expansion. It is related to the coupling of the Chern-Simons number to a massless Abelian gauge field. The result implies either that there is a way to distinguish between the phases, even though the macroscopic thermodynamical properties of the system have been observed to be smoothly connected, or that the perturbative Coleman-Hill theorem on which the argument is based, is circumvented by non-perturbative effects. We point out that this question could in principle be studied with three-dimensional lattice simulations.