- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

From BPS geodesics to mode-driven dynamics in the scattering of multiple BPS vortices

A. Alonso Izquierdo1,2, M. Bachmaier3,4, and A. Wereszczynski5,6

PDF

Phys. Rev. D 114, 015015 – Published 9 July, 2026

DOI: https://doi.org/10.1103/rpzh-fx62

Abstract

We analyze how the geodesic motion in the three- and four-vortex sectors of the Abelian-Higgs model at critical coupling is deformed by the excitation of a massive bound mode. We find that the geodesics corresponding to Bogomol’nyi-Prasad-Sommerfield (BPS) solutions with enhanced symmetry remain unchanged, although the direction of the actual motion depends on the mode-generated force, i.e., a force arising from the change of the mode frequency along the geodesic. In a generic case, for example, in head-on collisions between the axially symmetric one- and two-vortex or between two two-vortices, the vortex trajectories can differ strongly from the BPS geodesic. This enhances the chaotic behavior in the formation of the final state.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (60)

  1. N. S. Manton, A remark on the scattering of BPS monopoles, Phys. Lett. 110B, 54 (1982).
  2. T. M. Samols, Vortex scattering, Commun. Math. Phys. 145, 149 (1992).
  3. P. J. Ruback, Vortex string motion in the Abelian Higgs model, Nucl. Phys. B296, 669 (1988).
  4. E. P. S. Shellard and P. J. Ruback, Vortex scattering in two-dimensions, Phys. Lett. B 209, 262 (1988).
  5. Eric Myers, Claudio Rebbi, and Richard Strilka, A study of the interaction and scattering of vortices in the Abelian Higgs (or Ginzburg-Landau) model, Phys. Rev. D 45, 1355 (1992).
  6. Steffen Krusch, Morgan Rees, and Thomas Winyard, Scattering of vortices with excited normal modes, Phys. Rev. D 110, 056050 (2024).
  7. A. Alonso Izquierdo, N. S. Manton, J. Mateos Guilarte, and A. Wereszczynski, Collective coordinate models for 2-vortex shape mode dynamics, Phys. Rev. D 110, 085006 (2024).
  8. A. Alonso-Izquierdo, J. Mateos Guillarte, M. Rees, and A. Wereszczynski, Spectral wall in collisions of excited Abelian Higgs vortices, Phys. Rev. D 110, 065004 (2024).
  9. A. Alonso-Izquierdo, N. S. Manton, J. Mateos Guilarte, M. Rees, and A. Wereszczynski, Dynamics of excited BPS 3-vortices, Phys. Rev. D 111, 105021 (2025).
  10. https://youtu.be/o-0fQn-0wlE.
  11. Holger Bech Nielsen and P. Olesen, Vortex line models for dual strings, Nucl. Phys. B61, 45 (1973).
  12. E. B. Bogomolny, Stability of classical solutions, Sov. J. Nucl. Phys. 24, 449 (1976), https://inspirehep.net/literature/101280.
  13. Erick J. Weinberg, Multivortex solutions of the Ginzburg-Landau equations, Phys. Rev. D 19, 3008 (1979).
  14. Clifford Henry Taubes, Arbitrary N: Vortex solutions to the first order Landau-Ginzburg equations, Commun. Math. Phys. 72, 277 (1980).
  15. K. Arthur and J. Burzlaff, Existence theorems for pi / n vortex scattering, Lett. Math. Phys. 36, 311 (1996).
  16. A. Alonso-Izquierdo, W. Garcia Fuertes, N. S. Manton, and J. Mateos Guilarte, Spectral flow of vortex shape modes over the BPS 2-vortex moduli space, J. High Energy Phys. 01 (2024) 020.
  17. A. Alonso-Izquierdo and D. Miguelez-Caballero, Dissecting normal modes of vibration on vortices in Ginzburg-Landau superconductors, Phys. Rev. D 110, 125026 (2024).
  18. Michael Goodband and Mark Hindmarsh, Bound states and instabilities of vortices, Phys. Rev. D 52, 4621 (1995).
  19. C. Adam, K. Oles, T. Romanczukiewicz, and A. Wereszczynski, Spectral walls in soliton collisions, Phys. Rev. Lett. 122, 241601 (2019).
  20. J. I. Rawlinson, Coriolis terms in skyrmion quantization, Nucl. Phys. B949, 114800 (2019).
  21. D. Miguélez-Caballero, S. Navarro-Obregón, and A. Wereszczynski, Moduli space metric of the excited vortex, Phys. Rev. D 111, 105008 (2025).
  22. N. S. Manton, K. Oles, T. Romanczukiewicz, and A. Wereszczynski, Collective coordinate model of kink-antikink collisions in ϕ4 theory, Phys. Rev. Lett. 127, 071601 (2021).
  23. T. Sugiyama, Kink-antikink collisions in the two-dimensional phi**4 model, Prog. Theor. Phys. 61, 1550 (1979).
  24. David K. Campbell, Jonathan F. Schonfeld, and Charles A. Wingate, Resonance structure in kink-antikink interactions in φ 4 theory, Physica D (Amsterdam) 9, 1 (1983).
  25. M. Bachmaier and A. Wereszczynski, Resonance phenomena in vortex-antivortex collisions, Phys. Lett. B 875, 140324 (2026).
  26. Ta-Pei Cheng and Ling-Fong Li, Gauge Theory of Elementary Particle Physics (Oxford University Press, New York, 1984).
  27. Jose J. Blanco-Pillado, Daniel Jiménez-Aguilar, and Jon Urrestilla, Exciting the domain wall soliton, J. Cosmol. Astropart. Phys. 01 (2021) 027.
  28. Jose J. Blanco-Pillado, Daniel Jiménez-Aguilar, Jose M. Queiruga, and Jon Urrestilla, Internal excitations of global vortices, J. Cosmol. Astropart. Phys. 10 (2021) 047.
  29. Naoya Kitajima and Shota Nakagawa, Abelian-Higgs vortices in the oscillating axion background, Phys. Lett. B 871, 139993 (2025).
  30. N. S. Manton and H. Merabet, ϕ4 kinks—Gradient flow and dynamics, Nonlinearity 10, 3 (1997).
  31. A. Alonso-Izquierdo, J. J. Blanco-Pillado, D. Miguélez-Caballero, S. Navarro-Obregón, and J. Queiruga, Excited Abelian-Higgs vortices: Decay rate and radiation emission, Phys. Rev. D 110, 065009 (2024).
  32. Jose J. Blanco-Pillado, Daniel Jiménez-Aguilar, Jose M. Queiruga, and Jon Urrestilla, Parametric resonances in axionic cosmic strings, J. Cosmol. Astropart. Phys. 04 (2023) 043.
  33. T. W. B. Kibble, Topology of cosmic domains and strings, J. Phys. A 9, 1387 (1976).
  34. A. Vilenkin and A. E. Everett, Cosmic strings and domain walls in models with Goldstone and pseudo-Goldstone bosons, Phys. Rev. Lett. 48, 1867 (1982).
  35. Steven Weinberg, A new light boson?, Phys. Rev. Lett. 40, 223 (1978).
  36. Frank Wilczek, Problem of strong P and T invariance in the presence of instantons, Phys. Rev. Lett. 40, 279 (1978).
  37. John Preskill, Mark B. Wise, and Frank Wilczek, Cosmology of the invisible axion, Phys. Lett. 120B, 127 (1983).
  38. Alexander Vilenkin and Tanmay Vachaspati, Radiation of Goldstone bosons from cosmic strings, Phys. Rev. D 35, 1138 (1987).
  39. D. Garfinkle and T. Vachaspati, Fields due to kinky, cuspless, cosmic loops, Phys. Rev. D 37, 257 (1988).
  40. R. A. Battye and E. P. S. Shellard, Global string radiation, Nucl. Phys. B423, 260 (1994).
  41. R. A. Battye and E. P. S. Shellard, Radiative back reaction on global strings, Phys. Rev. D 53, 1811 (1996).
  42. Amelia Drew and E. P. S. Shellard, Radiation from global topological strings using adaptive mesh refinement: Methodology and massless modes, Phys. Rev. D 105, 063517 (2022).
  43. T. W. B. Kibble, George Lazarides, and Q. Shafi, Walls bounded by strings, Phys. Rev. D 26, 435 (1982).
  44. Gia Dvali, Lucy Komisel, and Anja Stuhlfauth, Cosmic strings and domain walls of the QCD quark condensate with and without a hidden axion, Phys. Rev. D 112, 096023 (2025).
  45. Josu C. Aurrekoetxea, Jose J. Blanco-Pillado, Alberto García Martín-Caro, and J. M. Queiruga, On curvature corrections for field theory cosmic strings, arXiv:2603.05243.
  46. Mark Hindmarsh, Joanes Lizarraga, Ander Urio, and Jon Urrestilla, Loop decay in Abelian-Higgs string networks, Phys. Rev. D 104, 043519 (2021).
  47. Jose J. Blanco-Pillado, Ken D. Olum, and Benjamin Shlaer, The number of cosmic string loops, Phys. Rev. D 89, 023512 (2014).
  48. Mark Hindmarsh, Joanes Lizarraga, Jon Urrestilla, David Daverio, and Martin Kunz, Scaling from gauge and scalar radiation in Abelian Higgs string networks, Phys. Rev. D 96, 023525 (2017).
  49. Jose J. Blanco-Pillado, Daniel Jiménez-Aguilar, Joanes Lizarraga, Asier Lopez-Eiguren, Ken D. Olum, Ander Urio, and Jon Urrestilla, Nambu-Goto dynamics of field theory cosmic string loops, J. Cosmol. Astropart. Phys. 05 (2023) 035.
  50. D. Stuart, Dynamics of Abelian Higgs vortices in the near Bogomolny regime, Commun. Math. Phys. 159, 51 (1994).
  51. J. M. Speight, Static intervortex forces, Phys. Rev. D 55, 3830 (1997).
  52. Martin Speight and Thomas Winyard, Short-range intervortex forces, Phys. Rev. D 112, 055024 (2025).
  53. Minoru Eto, Toshiaki Fujimori, Muneto Nitta, Keisuke Ohashi, and Norisuke Sakai, Dynamics of non-Abelian vortices, Phys. Rev. D 84, 125030 (2011).
  54. George Theodorou, Bruno Barton-Singer, and Stavros Komineas, Interaction and collision of skyrmions in chiral antiferromagnets, SciPost Phys. 18, 037 (2025).
  55. N. S. Manton, Unstable manifolds and soliton dynamics, Phys. Rev. Lett. 60, 1916 (1988).
  56. Maximilian Bachmaier, Gia Dvali, Josef Seitz, and Juan Sebastián Valbuena-Bermúdez, Simulations of magnetic monopole collisions, Phys. Rev. D 111, 075014 (2025).
  57. Gia Dvali, Swift memory burden in merging black holes: How information load affects black hole’s classical dynamics, arXiv:2509.22540.
  58. Daniel G. Figueroa, Adrien Florio, Francisco Torrenti, and Wessel Valkenburg, The art of simulating the early universe—part I, J. Cosmol. Astropart. Phys. 04 (2021) 035.
  59. Marcelo Gleiser and Andrew Sornborger, Longlived localized field configurations in small lattices: Application to oscillons, Phys. Rev. E 62, 1368 (2000).
  60. Siu Kwan Lam, Antoine Pitrou, and Stanley Seibert, Numba: A LLVM-based python JIT compiler, in Proceedings of the Second Workshop on the LLVM Compiler Infrastructure in HPC (Association for Computing Machinery, New York, 2015), pp. 1–6.

Outline

Information

Phys. Rev. D 114, 015015– Published 9 July, 2026
  • Received 24 March 2026
  • Accepted 26 June 2026
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation