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Vacuum energy and topological mass from a constant magnetic field and boundary conditions in coupled scalar field theories

A. J. D. Farias Junior1,*, Andrea Erdas2,†, and Herondy F. Santana Mota3,‡

  • *Contact author: antonio.farias@ifal.edu.br
  • †Contact author: aerdas@loyola.edu
  • ‡Contact author: hmota@fisica.ufpb.br
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Phys. Rev. D 112, 125026 – Published 22 December, 2025

DOI: https://doi.org/10.1103/qpkb-4rkl

Abstract

We investigate the combined effects of a uniform magnetic field and boundary conditions on vacuum energy and topological mass generation in a coupled scalar field theory. The system consists of a real scalar field, subject to Dirichlet boundary conditions, interacting via self- and cross-couplings with a gauge-coupled complex scalar field obeying mixed boundary conditions between two perfectly reflecting parallel plates. The magnetic field induces Landau quantization, leading to novel contributions. Employing zeta-function regularization within the effective potential formalism, we derive the renormalized effective potential up to second order in the coupling constants without imposing a vanishing magnetic field in the renormalization scheme. Our renormalization approach preserves magnetic contributions while properly removing divergences, enabling a consistent treatment of finite-size corrections, magnetic effects, and interaction terms. We compute the vacuum energy per unit area of the plates, analyze the emergence of a topological mass from boundary and magnetic contributions, and evaluate the first-order coupling-constant corrections at two-loop order. Detailed asymptotic analyses are presented for both weak- and strong-field regimes, revealing exponential suppression at high magnetic fields and nontrivial polynomial and logarithmic behavior in the weak-field limit.

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References (50)

  1. H. B. Casimir, On the attraction between two perfectly conducting plates, Proc. Kon. Ned. Akad. Wet. 51, 793 (1948).
  2. G. Bressi, G. Carugno, R. Onofrio, and G. Ruoso, Measurement of the Casimir force between parallel metallic surfaces, Phys. Rev. Lett. 88, 041804 (2002).
  3. M. J. Sparnaay, Measurements of attractive forces between flat plates, Physica 24, 751 (1958).
  4. S. K. Lamoreaux, Demonstration of the Casimir force in the 0.6 to 6  μm range, Phys. Rev. Lett. 78, 5 (1997).
  5. U. Mohideen and A. Roy, Precision measurement of the Casimir force from 0.1 to 0.9 micrometers, Phys. Rev. Lett. 81, 4549 (1998).
  6. W. J. Kim, M. Brown-Hayes, D. A. R. Dalvit, J. H. Brownell, and R. Onofrio, Anomalies in electrostatic calibrations for the measurement of the Casimir force in a sphere-plane geometry, Phys. Rev. A 78, 020101 (2008).
  7. Q. Wei, D. A. R. Dalvit, F. C. Lombardo, F. D. Mazzitelli, and R. Onofrio, Results from electrostatic calibrations for measuring the Casimir force in the cylinder-plane geometry, Phys. Rev. A 81, 052115 (2010).
  8. M. V. Cougo-Pinto, C. Farina, M. R. Negrao, and A. C. Tort, Bosonic Casimir effect in an external magnetic field, J. Phys. A 32, 4457 (1999).
  9. E. Elizalde, F. C. Santos, and A. C. Tort, Confined quantum fields under the influence of a uniform magnetic field, J. Phys. A 35, 7403 (2002).
  10. M. Ostrowski, Casimir effect in external magnetic field, Acta Phys. Pol. B 37, 1753 (2006).
  11. Y. A. Sitenko and S. A. Yushchenko, The Casimir effect with quantized charged scalar matter in background magnetic field, Int. J. Mod. Phys. A 29, 1450052 (2014).
  12. S. K. Blau, M. Visser, and A. Wipf, Analytic results for the effective action, Int. J. Mod. Phys. A 06, 5409 (1991).
  13. A. Erdas, Casimir effect of a Lorentz-violating scalar in magnetic field, Int. J. Mod. Phys. A 35, 2050209 (2020).
  14. A. Erdas, Casimir effect of a doubly Lorentz-violating scalar in magnetic field, Int. J. Mod. Phys. A 39, 2450141 (2024).
  15. A. Erdas, Magnetic Casimir effect of a Lorentz-violating scalar with higher-order derivatives, Int. J. Mod. Phys. A 40, 2550061 (2025).
  16. A. Erdas, Thermal effects on the Casimir energy of a Lorentz-violating scalar in magnetic field, Int. J. Mod. Phys. A 36, 20 (2021).
  17. A. Erdas and K. P. Seltzer, Finite temperature Casimir effect for charged massless scalars in a magnetic field, Phys. Rev. D 88, 105007 (2013).
  18. A. Erdas and K. P. Seltzer, Finite temperature Casimir effect for massive scalars in a magnetic field, Int. J. Mod. Phys. A 29, 1450091 (2014).
  19. A. Erdas, Magnetic field corrections to the repulsive Casimir effect at finite temperature, Int. J. Mod. Phys. A 31, 07 (2016).
  20. A. Erdas, Finite temperature Casimir effect of a Lorentz-violating scalar with higher order derivatives, Eur. Phys. J. C 85, 1391 (2025).
  21. M. V. Cougo-Pinto, C. Farina, and A. C. Tort, Fermionic Casimir effect in an external magnetic field, Conf. Proc. C 9809142, 235 (1999).
  22. Y. A. Sitenko, Influence of quantized massive matter fields on the Casimir effect, Mod. Phys. Lett. A 30, 1550099 (2015).
  23. A. Erdas, Finite temperature Casimir effect for massless Majorana fermions in a magnetic field, Phys. Rev. D 83, 025005 (2011).
  24. A. Erdas, Magnetic corrections to the fermionic Casimir effect in Horava-Lifshitz theories, Int. J. Mod. Phys. A 38, 2350117 (2023).
  25. A. Erdas and G. Feldman, Magnetic field effects on Lagrangians and neutrino self-energies in the Salam-Weinberg theory in arbitrary gauges, Nucl. Phys. B343, 597 (1990).
  26. M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Advances in the Casimir effect (OUP, Oxford, 2009), Vol. 145.
  27. E. Elizalde, S. D. Odintsov, A. Romeo, A. A. Bytsenko, and S. Zerbini, Zeta Regularization Techniques with Applications (World Scientific Publishing, Singapore, 1994).
  28. K. A. Milton, The Casimir Effect: Physical Manifestations of Zero-Point Energy (World Scientific, Singapore, 2001).
  29. D. J. Toms, Interacting twisted and untwisted scalar fields in a nonsimply connected space-time, Ann. Phys. (N.Y.) 129, 334 (1980).
  30. D. J. Toms, Symmetry breaking and mass generation by space-time topology, Phys. Rev. D 21, 2805 (1980).
  31. P. J. Porfírio, H. F. Santana Mota, and G. Q. Garcia, Ground state energy and topological mass in spacetimes with nontrivial topology, Int. J. Mod. Phys. D 30, 2150056 (2021).
  32. M. B. Cruz, E. R. Bezerra de Mello, and H. F. Santana Mota, Casimir energy and topological mass for a massive scalar field with Lorentz violation, Phys. Rev. D 102, 045006 (2020).
  33. A. J. D. Farias Junior and H. F. Mota Santana, Loop correction to the scalar Casimir energy density and generation of topological mass due to a helix boundary condition in a scenario with Lorentz violation, Int. J. Mod. Phys. D 31, 2250126 (2022).
  34. A. J. D. F. Junior and H. F. S. Mota, Casimir effect, loop corrections, and topological mass generation for interacting real and complex scalar fields in Minkowski spacetime with different conditions, Phys. Rev. D 107, 125019 (2023).
  35. A. J. D. F. Junior, A. Smirnov, H. F. Santana Mota, and E. R. B. de Mello, Vacuum energy density for interacting real and complex scalar fields in a Lorentz symmetry violation scenario, Int. J. Mod. Phys. D 34, 2450069 (2025).
  36. A. Flachi, M. Nitta, S. Takada, and R. Yoshii, Fermion Casimir effect and magnetic Larkin-Ovchinnikov phases, Phys. Rev. D 111, 016003 (2025).
  37. Quantum field theory under the influence of external conditions, Proceedings, 9th Conference, QFEXT09, Norman, USA, 2009, edited by K. A. Milton and M. Bordag (2010), Vol. 25, 10.1142/7529.
  38. N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 1982).
  39. A. Altland and B. D. Simons, Condensed Matter Field Theory, 2nd ed. (Cambridge University Press, Cambridge, England, 2010).
  40. M. D. Schwartz, Quantum Field Theory and the Standard Model (Cambridge University Press, Cambridge, England, 2014).
  41. L. H. Ryder, Quantum Field Theory (Cambridge University Press, Cambridge, England, 1996).
  42. W. Greiner and J. Reinhardt, Field Quantization (Springer Science & Business Media, New York, 2013).
  43. E. Elizalde, Ten Physical Applications of Spectral Zeta Functions, Lecture Notes in Physics Monographs Vol. 35 (Springer-Verlag, Berlin, Heidelberg, 1995).
  44. V. Mostepanenko, N. Trunov, and R. Znajek, The Casimir Effect and Its Applications, Oxford Science Publications (Clarendon Press, New York, 1997).
  45. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series Vol. 55 (Dover Publications, New York, 1965).
  46. E. Elizalde, Ten Physical Applications of Spectral Zeta Functions, Lecture Notes in Physics Monographs Vol. 35 (Springer-Verlag, Berlin, Heidelberg, 1995).
  47. E. Elizalde, Explicit analytical continuation of the inhomogeneous epstein zeta function, Hiroshima University Preprint (1994).
  48. G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists: A Comprehensive Guide (Academic Press, New York, 2011).
  49. E. Elizalde, Ten Physical Applications of Spectral Zeta Functions (Springer, New York, 2012), Vol. 855.
  50. G. E. Andrews, R. Askey, and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications Vol. 71 (Cambridge University Press, Cambridge, England, 1999).

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Phys. Rev. D 112, 125026– Published 22 December, 2025
  • Received 21 October 2025
  • Accepted 9 December 2025
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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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