  To obtain the one-loop corrections to the mass of a kink by mode
regularization, one may take one-half the result for the mass of a widely
separated kink-antikink (or sphaleron) system, where the two bosonic zero modes
count as two degrees of freedom, but the two fermionic zero modes as only one
degree of freedom in the sums over modes. For a single kink, there is one
bosonic zero mode degree of freedom, but it is necessary to average over four
sets of fermionic boundary conditions in order (i) to preserve the fermionic
Z$_2$ gauge invariance $\psi \to -\psi$, (ii) to satisfy the basic principle of
mode regularization that the boundary conditions in the trivial and the kink
sector should be the same, (iii) in order that the energy stored at the
boundaries cancels and (iv) to avoid obtaining a finite, uniformly distributed
energy which would violate cluster decomposition. The average number of
fermionic zero-energy degrees of freedom in the presence of the kink is then
indeed 1/2. For boundary conditions leading to only one fermionic zero-energy
solution, the Z$_2$ gauge invariance identifies two seemingly distinct `vacua'
as the same physical ground state, and the single fermionic zero-energy
solution does not correspond to a degree of freedom. Other boundary conditions
lead to two spatially separated $\omega \sim 0 $ solutions, corresponding to
one (spatially delocalized) degree of freedom. This nonlocality is consistent
with the principle of cluster decomposition for correlators of observables.
