No saturation of the quantum Bogomolnyi bound by two-dimensional supersymmetric solitons

TL;DR

This paper reexamines the quantum mass correction of supersymmetric solitons in 2D models, arguing that proper mode counting shows the quantum Bogomolnyi bound is not saturated, contrary to previous claims.

Contribution

It demonstrates that only the mode cut-off method, with equal bosonic and fermionic modes, yields correct quantum mass corrections, challenging prior results on bound saturation.

Findings

Mode cut-off method is correct for quantum mass calculations.

Quantum Bogomolnyi bound is not saturated in 2D supersymmetric models.

Results align with exact spectrum calculations for sine-Gordon model.

Abstract

We reanalyse the question whether the quantum Bogomolnyi bound is saturated in the two-dimensional supersymmetric kink and sine-Gordon models. Our starting point is the usual expression for the one-loop correction to the mass of a soliton in terms of sums over zero-point energies. To regulate these sums, most authors put the system in a box with suitable boundary conditions, and impose an ultraviolet cut-off. We distinguish between an energy cut-off and a mode number cut-off, and show that they lead to different results. We claim that only the mode cut-off yields correct results, and only if one considers exactly the same number of bosonic and fermionic modes in the total sum over bound-state and zero-point energies. To substantiate this claim, we show that in the sine-Gordon model only the mode cut-off yields a result for the quantum soliton mass that is consistent with the exact result for the spectrum as obtained by Dashen et al. from quantising the so-called breather solution. In the supersymmetric case, our conclusion is that contrary to previous claims the quantum Bogomolnyi bound is not saturated in any of the two-dimensional models considered.