Mode regularization of the supersymmetric sphaleron and kink: Zero modes and discrete gauge symmetry - art. no. 045013

To obtain the one-loop corrections to the mass of a kink by mode regulariza tion. one may take one-half the result for the mass of a widely separated k ink-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 deg ree of freedom in the sums over modes. For a single kink, there is one boso nic zero mode degree of freedom, but it is necessary to average over four s ets of fermionic boundary conditions in order (i) to preserve the fermionic Z(2) gauge invariance psi--> - psi, (ii) to satisfy the basic principle of mode regularization that the boundary conditions in the trivial and the ki nk sector should be the same. (iii) that the energy stored at the boundarie s cancels and (iv) to avoid obtaining a finite, uniformly distributed energ y which would violate cluster decomposition. The average number of fermioni c 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 sol ution, the Z(2) gauge invariance identifies two seemingly distinct "vacua" as the same physical ground state, and the single fermionic zero-energy sol ution does not correspond to a degree of freedom. Other boundary conditions lead to two spatially separated omega similar to0 solutions, corresponding to one (spatially delocalized) degree of freedom. This nonlocality is cons istent with the principle of cluster decomposition for correlators of obser vables.