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Witten effect in the three-form description of θ vacua

Maximilian Bachmaier1,2,*, Gia Dvali1,2, and Juan Sebastián Valbuena-Bermúdez3,†

  • *Contact author: maximilian.bachmaier@physik.uni-muenchen.de
  • †Contact author: jvalbuena@ifae.es
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Phys. Rev. D 113, 095001 – Published 1 May, 2026

DOI: https://doi.org/10.1103/2mw1-vpb1

Abstract

The θ vacua of a gauge theory admit an equivalent formulation as vacua of a massless Chern-Simons three-form, which originate from the topological susceptibility of the vacuum. This formulation provides a framework in which the physical manifestations of the θ angle, which are quantum in origin, can be captured at the level of effective classical equations of motion. Within this framework, we derive the Witten effect, demonstrating that in the background of a massless three-form, the magnetic monopole indeed acquires an electric charge proportional to θ. This result, in particular, provides evidence that instantons, even when constrained by the Higgs effect, maintain a nonzero topological susceptibility of the vacuum. In addition to the Witten effect, we numerically demonstrate that a magnetic monopole exhibits polarizability when placed in a constant background electric field.

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Phys. Rev. D 113, 095001– Published 1 May, 2026
  • Received 24 February 2026
  • Accepted 8 April 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

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