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Topological boundary conditions, the BPS bound, and elimination of ambiguities in the quantum mass of solitons

Horatiu Nastase, Misha Stephanov, Peter van Nieuwenhuizen, Anton Rebhan

TL;DR

This work introduces topological boundary conditions and a mass-renormalization principle to fix UV ambiguities in the quantum mass of solitons in 1+1D SUSY theories. By treating the soliton mass as a vacuum-energy difference between topologically distinct sectors and enforcing topology-independence at the conformal point $m\to0$, the authors derive unambiguous one-loop results that agree with exact S-matrix data for $N=1$ SUSY sine-Gordon but reveal a non-saturation of the BPS bound, while $N=2$ theories show no quantum corrections and exact BPS saturation. The paper also develops a rigorous framework for the quantum Hamiltonian of solitons with collective coordinates, and provides a complete two-loop calculation in the sine-Gordon model demonstrating the absence of ambiguities at this order. Together, these results clarify the interplay between quantum soliton masses, central charges, and BPS bounds, and connect nonperturbative boundary conditions with perturbative regularization schemes and exact integrability data.

Abstract

We fix the long-standing ambiguity in the 1-loop contribution to the mass of a 1+1-dimensional supersymmetric soliton by adopting a set of boundary conditions which follow from the symmetries of the action and which depend only on the topology of the sector considered, and by invoking a physical principle that ought to hold generally in quantum field theories with a topological sector: for vanishing mass and other dimensionful constants, the vacuum energies in the trivial and topological sectors have to become equal. In the two-dimensional N=1 supersymmetric case we find a result which for the supersymmetric sine-Gordon model agrees with the known exact solution of the S-matrix but seems to violate the BPS bound. We analyze the nontrivial relation between the quantum soliton mass and the quantum BPS bound and find a resolution. For N=2 supersymmetric theories, there are no one-loop corrections to the soliton mass and to the central charge (and also no ambiguities) so that the BPS bound is always saturated. Beyond 1-loop there are no ambiguities in any theory, which we explicitly check by a 2-loop calculation in the sine-Gordon model.

Topological boundary conditions, the BPS bound, and elimination of ambiguities in the quantum mass of solitons

TL;DR

This work introduces topological boundary conditions and a mass-renormalization principle to fix UV ambiguities in the quantum mass of solitons in 1+1D SUSY theories. By treating the soliton mass as a vacuum-energy difference between topologically distinct sectors and enforcing topology-independence at the conformal point , the authors derive unambiguous one-loop results that agree with exact S-matrix data for SUSY sine-Gordon but reveal a non-saturation of the BPS bound, while theories show no quantum corrections and exact BPS saturation. The paper also develops a rigorous framework for the quantum Hamiltonian of solitons with collective coordinates, and provides a complete two-loop calculation in the sine-Gordon model demonstrating the absence of ambiguities at this order. Together, these results clarify the interplay between quantum soliton masses, central charges, and BPS bounds, and connect nonperturbative boundary conditions with perturbative regularization schemes and exact integrability data.

Abstract

We fix the long-standing ambiguity in the 1-loop contribution to the mass of a 1+1-dimensional supersymmetric soliton by adopting a set of boundary conditions which follow from the symmetries of the action and which depend only on the topology of the sector considered, and by invoking a physical principle that ought to hold generally in quantum field theories with a topological sector: for vanishing mass and other dimensionful constants, the vacuum energies in the trivial and topological sectors have to become equal. In the two-dimensional N=1 supersymmetric case we find a result which for the supersymmetric sine-Gordon model agrees with the known exact solution of the S-matrix but seems to violate the BPS bound. We analyze the nontrivial relation between the quantum soliton mass and the quantum BPS bound and find a resolution. For N=2 supersymmetric theories, there are no one-loop corrections to the soliton mass and to the central charge (and also no ambiguities) so that the BPS bound is always saturated. Beyond 1-loop there are no ambiguities in any theory, which we explicitly check by a 2-loop calculation in the sine-Gordon model.

Paper Structure

This paper contains 9 sections, 142 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Eliminating the one-loop ultraviolet ambiguity using a physical principle
  3. Fermions, supersymmetry, and topological boundary conditions
  4. Quantum BPS bound, soliton mass, and finite size effects
  5. The N=2 case
  6. Higher loops
  7. The quantum Hamiltonian
  8. The actual two-loop calculation
  9. Conclusions

Figures (2)

  • Figure 1: The left- and the right-hand sides of the equation $\delta(k) = 2\pi n + \pi - kL$ are plotted schematically by solid lines in the case of the $\phi^4$ kink (two bound states). The dashed line represents the value of $\delta(k)$ without the discontinuity $2\pi\varepsilon(k)$. Observe that with this discontinuity the mode numbers $n=-1,0$ should be left out, while the spectrum of allowed values of $k$ is not affected.
  • Figure 2: The left- and the right-hand sides of the equation $\delta + \theta/2 = 2\pi n + \pi - kL$ are plotted schematically in the case of the supersymmetric $\phi^4$ kink (two bound states). The dashed line represent the value of $\delta(k)+\theta(k)/2$ without the discontinuity $2\pi\varepsilon(k)$. As in the bosonic spectrum the discontinuity leads to $n=-1,0$ mode numbers being skipped, while the spectrum of allowed values of $k$ is not affected.