Mode regularization of the susy sphaleron and kink: zero modes and discrete gauge symmetry

TL;DR

This paper investigates mode regularization for calculating one-loop mass corrections of kinks and sphalerons in supersymmetric theories, emphasizing zero mode treatment, boundary conditions, and gauge invariance to ensure consistent physical results.

Contribution

It introduces a method for averaging over boundary conditions to correctly account for fermionic zero modes and gauge invariance in supersymmetric kink and sphaleron calculations.

Findings

Fermionic zero modes are effectively half a degree of freedom after averaging.

Boundary conditions influence the count of zero modes and gauge invariance.

Nonlocal zero modes are consistent with cluster decomposition principles.

Abstract

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.