The level-crossing of the fermion system described by the Euclidean Dirac Hamiltonian in the valley background, is studied in non-Abelian gauge theories. A chiral fermion level is shown to cross twice zero in the case of well-separated instanton-anti-instanton background. Below a critical separation, however, level-crossing is found not to occur. The phenomenon can be interpreted as the transition to a gauge-field configuration of purely perturbative nature, below a critical instanton-anti-instanton separation. In the context of high-energy electroweak interactions, our findings seem to definitely invalidate some optimistic argument concerning the observability of baryon number violation based on the use of the optical theorem in conjunction with the valley fields.
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