- 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

Probing the vacuum as a chiral medium

T. Heinzl*, B. King†, and A. Mercuri-Baron‡

  • *Contact author: t.heinzl@plymouth.ac.uk
  • †Contact author: b.king@plymouth.ac.uk
  • ‡Contact author: anthony.mercuri-baron@plymouth.ac.uk
PDF

Phys. Rev. D 113, 036031 – Published 27 February, 2026

DOI: https://doi.org/10.1103/8h6n-fm46

Abstract

We study the circular birefringence experienced by linearly polarized photons colliding with a circularly polarized background creating a vacuum of definite chirality (handedness). For this scenario the standard Heisenberg-Euler approach fails and must be supplemented by derivative corrections which we match to known Hilbert series. Choosing a plane wave background, we find equivalence in the low-energy limit between three approaches: (i) adding derivative corrections to the Heisenberg-Euler Lagrangian, (ii) improving the locally constant field approximation to the one-loop polarization tensor, and (iii) performing a low-energy expansion of the direct 2→2 QED photon-photon scattering amplitude. Going beyond plane-wave backgrounds, we analyze an example of a circularly polarized standing wave sensitive to derivative corrections. We find a parameter regime where these corrections could be probed in experiments.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (82)

  1. N. Ahmadiniaz et al., High Power Laser Sci. Eng. 13, e7 (2025).
  2. R. P. Mignani, V. Testa, D. G. Caniulef, R. Taverna, R. Turolla, S. Zane, and K. Wu, Mon. Not. R. Astron. Soc. 465, 492 (2017).
  3. D. d’Enterria and G. G. da Silveira, Phys. Rev. Lett. 111, 080405 (2013); 116, 129901(E) (2016).
  4. M. Aaboud et al. (ATLAS Collaboration), Nat. Phys. 13, 852 (2017).
  5. A. M. Sirunyan et al. (CMS Collaboration), Phys. Lett. B 797, 134826 (2019).
  6. G. Aad et al. (ATLAS Collaboration), Phys. Rev. Lett. 123, 052001 (2019).
  7. J. D. Brandenburg, J. Seger, Z. Xu, and W. Zha, Rep. Prog. Phys. 86, 083901 (2023).
  8. A. M. Sirunyan et al. (CMS Collaboration), Phys. Lett. B 797, 134826 (2019).
  9. J. Agil, R. Battesti, and C. Rizzo, Eur. Phys. J. D 75, 90 (2021).
  10. A. Ejlli, F. Della Valle, U. Gastaldi, G. Messineo, R. Pengo, G. Ruoso, and G. Zavattini, Phys. Rep. 871, 1 (2020).
  11. H.-P. Schlenvoigt, T. Heinzl, U. Schramm, T. E. Cowan, and R. Sauerbrey, Phys. Scr. 91, 023010 (2016).
  12. H. G. Rinderknecht et al., Phys. Plasmas 32, 083301 (2025).
  13. T. Heinzl, B. King, and D. Liu, Phys. Rev. D 111, 056018 (2025).
  14. M. Marklund and P. K. Shukla, Rev. Mod. Phys. 78, 591 (2006).
  15. A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, Rev. Mod. Phys. 84, 1177 (2012).
  16. B. King and T. Heinzl, High Power Laser Sci. Eng. 4, e5 (2016).
  17. A. Fedotov, A. Ilderton, F. Karbstein, B. King, D. Seipt, H. Taya, and G. Torgrimsson, Phys. Rep. 1010, 1 (2023).
  18. G. Sarri et al., Eur. Phys. J. Plus 140, 1151 (2025).
  19. I. d. M. Froldi and F. L. Braghin, Int. J. Mod. Phys. A 40, 2550116 (2025).
  20. I. Affleck, J. Phys. A 21, 693 (1988).
  21. B. King, T. Heinzl, and T. G. Blackburn, Eur. Phys. J. C 83, 901 (2023).
  22. W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936).
  23. J. M. Davila, C. Schubert, and M. A. Trejo, Int. J. Mod. Phys. A 29, 1450174 (2014).
  24. H. Euler and B. Kockel, Naturwissenschaften 23, 246 (1935).
  25. P. A. M. Dirac, in Stucture et Propriétés des Noyaux Atomiques (Gauthier-Villars, Cambridge, England, 1934), pp. 203–230; Proceedings of the 7th Solvay Congress, Brussels, 1933 (Republished in Early Quantum Electrodynamics, Cambridge UP, 1995), pp. 136–144.
  26. W. Heisenberg, Z. Phys. 90, 209 (1934); 92, 692(E) (1934). W. Heisenberg [English translation: Remarks on the Dirac theory of the positron, in A. I. Miller, Early Quantum Electrodynamics (Cambridge, UP, 1995), pp. 169–187].
  27. E. A. Uehling, Phys. Rev. 48, 55 (1935).
  28. X.-w. Kong and F. Ravndal, Nucl. Phys. B526, 627 (1998).
  29. F. Bopp, Ann. Phys. (Leipzig) 430, 345 (1940).
  30. B. Podolsky, Phys. Rev. 62, 68 (1942).
  31. A. E. Zayats, Ann. Phys. (Amsterdam) 342, 11 (2014).
  32. J. Gratus, V. Perlick, and R. W. Tucker, J. Phys. A 48, 435401 (2015).
  33. D. A. Dicus, C. Kao, and W. W. Repko, Phys. Rev. D 57, 2443 (1998).
  34. V. P. Gusynin and I. A. Shovkovy, J. Math. Phys. (N.Y.) 40, 5406 (1999).
  35. F. Karbstein, J. High Energy Phys. 09 (2021) 070.
  36. B. De Tollis, Nuovo Cimento 32, 757 (1964).
  37. L. Lehman and A. Martin, Phys. Rev. D 91, 105014 (2015).
  38. L. Lehman and A. Martin, J. High Energy Phys. 02 (2016) 081.
  39. B. Henning, X. Lu, T. Melia, and H. Murayama, J. High Energy Phys. 08 (2017) 016; 09 (2019) 019(E).
  40. B. Henning, X. Lu, T. Melia, and H. Murayama, Commun. Math. Phys. 347, 363 (2016).
  41. B. Henning, X. Lu, T. Melia, and H. Murayama, J. High Energy Phys. 10 (2017) 199.
  42. M. Ruhdorfer, J. Serra, and A. Weiler, J. High Energy Phys. 05 (2020) 083.
  43. L. Silberstein, Ann. Phys. 327, 579 (1907).
  44. H. Stephani, Relativity (Cambridge University Press, Cambridge, England, 2004).
  45. K. Colwell and J. Terning, J. High Energy Phys. 03 (2016) 068.
  46. L. C. Martin, C. Schubert, and V. M. Villanueva Sandoval, Nucl. Phys. B668, 335 (2003).
  47. B. De Tollis, Nuovo Cimento 35, 1182 (1965).
  48. A. H. Ajjath, E. Chaubey, and H.-S. Shao, J. High Energy Phys. 03 (2024) 121.
  49. A. H. Ajjath, E. Chaubey, M. Fraaije, V. Hirschi, and H.-S. Shao, Phys. Lett. B 851, 138555 (2024).
  50. G. Baym, Lectures on Quantum Mechanics (Westview Press, Boulder, CO, 1990).
  51. P. H. Damgaard, L. Plante, and P. Vanhove, J. High Energy Phys. 11 (2021) 213.
  52. P. H. Damgaard, E. R. Hansen, L. Planté, and P. Vanhove, J. High Energy Phys. 09 (2023) 183.
  53. D. Meschede, Optics, Light and Lasers (Wiley-VCH, Weinheim, 2004).
  54. J. Peatross and M. Ware, Physics of Light and Optics (Brigham Young University, Provo, UT, 2025), revised edition, available at https://optics.byu.edu.
  55. T. Heinzl, A. Ilderton, and B. King, Phys. Rev. Lett. 127, 061601 (2021).
  56. G. Torgrimsson, Phys. Rev. Lett. 127, 111602 (2021).
  57. F. Karbstein, D. Ullmann, E. A. Mosman, and M. Zepf, Phys. Rev. Lett. 129, 061802 (2022).
  58. Z. Wang, C. Feng, T. Winsor, and Y. Liu, Nanotechnology 27, 412001 (2016).
  59. H. Poincare, Théorie mathématique de la lumière (George Carré, Paris, 1892), Vol. II.
  60. Francois Ernest Mallard, Traité de cristallographie géométrique et physique (Dunod, Paris, 1884), Vol. II.
  61. D. Ives, Polarisation Effects and Measurements in Optical Fibre Systems (National Physical Laboratory, Teddington, UK, 2004), Measurement Good Practice Guide No. 67.
  62. A. Zangwill, Modern Electrodynamics (Cambridge University Press, Cambridge, England, 2012).
  63. M. R. Dennis and M. A. Alonso, Phil. Trans. R. Soc. A 375, 20150441 (2017).
  64. T. Heinzl and A. Ilderton, Phys. Rev. Lett. 118, 113202 (2017).
  65. N. B. Narozhnyĭ, Sov. Phys. JETP 28, 371 (1969).
  66. V. N. Baĭer, A. I. Mil’shteĭn, and V. M. Strakhovenko, Sov. Phys. JETP 42, 961 (1976).
  67. S. Meuren, C. H. Keitel, and A. Di Piazza, Phys. Rev. D 88, 013007 (2013).
  68. V. Dinu, T. Heinzl, A. Ilderton, M. Marklund, and G. Torgrimsson, Phys. Rev. D 89, 125003 (2014).
  69. A. Di Piazza, M. Tamburini, S. Meuren, and C. H. Keitel, Phys. Rev. A 99, 022125 (2019).
  70. A. Ilderton, B. King, and D. Seipt, Phys. Rev. A 99, 042121 (2019).
  71. B. King, Phys. Rev. A 101, 042508 (2020).
  72. F. W. J. Olver, Asymptotics and Special Functions (AKP Classics, A K Peters Ltd., Natick, 1997).
  73. NIST, Nist Digital Library of Mathematical Functions, http://dlmf.nist.gov/ (2025).
  74. I. A. Aleksandrov and V. M. Shabaev, arXiv:2303.16273.
  75. E. Lundström, G. Brodin, J. Lundin, M. Marklund, R. Bingham, J. Collier, J. T. Mendonça, and P. Norreys, Phys. Rev. Lett. 96, 083602 (2006).
  76. H. Gies, F. Karbstein, C. Kohlfürst, and N. Seegert, Phys. Rev. D 97, 076002 (2018).
  77. R. Aboushelbaya et al., Phys. Rev. Lett. 123, 113604 (2019).
  78. A. J. Macleod and B. King, Phys. Rev. A 110, 032216 (2024).
  79. I. Agapov et al., Technical Report, CERN, Geneva, 2025, https://cds.cern.ch/record/2928809.
  80. A. J. Macleod, J. P. Edwards, T. Heinzl, B. King, and S. V. Bulanov, New J. Phys. 25, 093002 (2023).
  81. H. Abramowicz et al., Eur. Phys. J. Special Topics 230, 2445 (2021).
  82. H. Abramowicz et al. (LUXE Collaboration), Eur. Phys. J. Special Topics 233, 1709 (2024).

Outline

Information

Phys. Rev. D 113, 036031– Published 27 February, 2026
  • Received 18 November 2025
  • Accepted 5 February 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