Winding transitions at finite energy and temperature: An O(3) model

Winding number transitions in the two-dimensional softly broken O(3) nonlinear $\sigma$ model are studied at finite energy and temperature. New periodic instanton solutions which dominate the semiclassical transition amplitudes are found analytically at low energies, and numerically for all energies up to the sphaleron scale. The Euclidean period $\beta$ of these finite energy instantons increases with energy, contrary to the behavior found in the Abelian Higgs model or simple one-dimensional systems. This results in a sharp crossover from instanton-dominated tunneling to sphaleron-dominated thermal activation at a certain critical temperature. Since this behavior is traceable to the soft breaking of conformal invariance by the mass term in the $\sigma$ model, semiclassical winding number transition amplitudes in the electroweak theory in 3+1 dimensions should exhibit a similar sharp crossover. We argue that this is indeed the case in the standard model for $M_H<4\mathrm{}{M}_W$.