Decoupling in Non-Perturbative Background Fields: the Thermal Sphaleron

Standard decoupling of heavy fermions may fail when there are non-perturbative variations in a scalar field which gives masses to the fermions. One situation of phenomenological relevance is the case of sphalerons in the presence of fermions at finite temperatures. The free energy of a simple model is determined using a non-perturbative technique to study the effect of fermions on the scalar field. The effects of quantum and thermal fermionic fluctuations on the free energy of the thermal sphaleron are calculated, including contributions from the gradients of the scalar field to all orders.