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Standard decoupling of heavy fermions may fail when there are nonperturbative 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 external fermions at finite temperatures. The free energy of a simple model is determined using a nonperturbative technique to study the effect of fermions on the scalar field. The effects of quantum and thermal fermion fluctuations on the free energy of the sphaleron are calculated, including all orders in gradients of the scalar field.
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Gould et al. (1993) studied this question.
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