- 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

Area law for entanglement entropy in particle scattering

Ian Low and Zhewei Yin

PDF

Phys. Rev. D 113, 065004 – Published 6 March, 2026

DOI: https://doi.org/10.1103/3yg7-r5s9

Abstract

The scattering cross section is the effective area of collision when two particles collide. Quantum mechanically, it is a measure of the probability for a specific process to take place. Employing wave packets to describe the scattering process, we compute the entanglement entropy in 2-to-2 scattering of particles in a general setting using the S-matrix formalism. Applying the optical theorem, we show that the linear entropy E2 is given by the elastic cross section σel in unit of the transverse size L2 of the wave packet, E2∼σel/L2, when the initial states are not entangled. The result allows for dual interpretations of the entanglement entropy as an area and as a probability. Since σel is generally believed, and observed experimentally, to grow with the collision energy s in the high energy regime, the result suggests a “second law” of entanglement entropy for high energy collisions. Furthermore, the Froissart bound places an upper limit on the entropy growth.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (51)

  1. C. E. Shannon, Bell Syst. Tech. J. 27, 379 (1948).
  2. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, England, 2012).
  3. E. Witten, Riv. Nuovo Cimento 43, 187 (2020).
  4. A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, Rev. Mod. Phys. 93, 035002 (2021).
  5. A. Rényi, On measures of entropy and information, in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability (University of California Press, Berkeley, 1960), pp. 547–562.
  6. C. Tsallis, J. Stat. Phys. 52, 479 (1988).
  7. J. D. Bekenstein, Phys. Rev. D 7, 2333 (1973).
  8. S. W. Hawking, Commun. Math. Phys. 43, 199 (1975); 46, 206(E) (1976).
  9. M. Srednicki, Phys. Rev. Lett. 71, 666 (1993).
  10. S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, 181602 (2006).
  11. J. Eisert, M. Cramer, and M. B. Plenio, Rev. Mod. Phys. 82, 277 (2010).
  12. G. Dvali, Phys. Rev. D 97, 105005 (2018).
  13. J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998).
  14. G. Dvali, C. Gomez, R. S. Isermann, D. Lüst, and S. Stieberger, Nucl. Phys. B893, 187 (2015).
  15. S. Seki, I. Y. Park, and S.-J. Sin, Phys. Lett. B 743, 147 (2015).
  16. R. Peschanski and S. Seki, Phys. Lett. B 758, 89 (2016).
  17. A. Cervera-Lierta, J. I. Latorre, J. Rojo, and L. Rottoli, SciPost Phys. 3, 036 (2017).
  18. J. Fan and X. Li, Phys. Rev. D 97, 016011 (2018).
  19. S. R. Beane, D. B. Kaplan, N. Klco, and M. J. Savage, Phys. Rev. Lett. 122, 102001 (2019).
  20. T. N. Tomaras and N. Toumbas, Phys. Rev. D 101, 065006 (2020).
  21. G. Dvali, J. High Energy Phys. 03 (2021) 126.
  22. R. Peschanski and S. Seki, Phys. Rev. D 100, 076012 (2019).
  23. R. Aoude, M.-Z. Chung, Y.-t. Huang, C. S. Machado, and M.-K. Tam, Phys. Rev. Lett. 125, 181602 (2020).
  24. I. Low and T. Mehen, Phys. Rev. D 104, 074014 (2021).
  25. G. Dvali and R. Venugopalan, Phys. Rev. D 105, 056026 (2022).
  26. G. Dvali and O. Sakhelashvili, Phys. Rev. D 105, 065014 (2022).
  27. B. Müller and A. Schäfer, arXiv:2211.16265.
  28. S. Fedida and A. Serafini, Phys. Rev. D 107, 116007 (2023).
  29. Q. Liu, I. Low, and T. Mehen, Phys. Rev. C 107, 025204 (2023).
  30. C. Cheung, T. He, and A. Sivaramakrishnan, Phys. Rev. D 108, 045013 (2023).
  31. M. Hentschinski, D. E. Kharzeev, K. Kutak, and Z. Tu, Phys. Rev. Lett. 131, 241901 (2023).
  32. R. A. Morales, Eur. Phys. J. Plus 138, 1157 (2023).
  33. M. Carena, I. Low, C. E. M. Wagner, and M.-L. Xiao, Phys. Rev. D 109, L051901 (2024).
  34. K. Sakurai and M. Spannowsky, Phys. Rev. Lett. 132, 151602 (2024).
  35. T.-R. Hu, S. Chen, and F.-K. Guo, Phys. Rev. D 110, 014001 (2024).
  36. R. Aoude, G. Elor, G. N. Remmen, and O. Sumensari, arXiv:2402.16956.
  37. K. Kowalska and E. M. Sessolo, J. High Energy Phys. 07 (2024) 156.
  38. H. Cheng and T. T. Wu, Phys. Rev. 186, 1611 (1969).
  39. H. Cheng and T. T. Wu, Phys. Rev. Lett. 24, 1456 (1970).
  40. M. M. Block and R. N. Cahn, Rev. Mod. Phys. 57, 563 (1985).
  41. G. Pancheri and Y. N. Srivastava, Eur. Phys. J. C 77, 150 (2017).
  42. M. Froissart, Phys. Rev. 123, 1053 (1961).
  43. A. Martin, Phys. Rev. 129, 1432 (1963).
  44. G. Lüders, Ann. Phys. (Berlin) 443, 322 (1950).
  45. I. Low and Z. Yin, Phys. Rev. D 111, 065027 (2025).
  46. H. Cheng and T. T. Wu, Phys. Rev. Lett. 22, 666 (1969).
  47. I. M. Dremin, Phys. Usp. 56, 3 (2013).
  48. R. L. Workman et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2022, 083C01 (2022).
  49. J. Chen, T. Han, and B. Tweedie, J. High Energy Phys. 11 (2017) 093.
  50. T. Han, Y. Ma, and K. Xie, Phys. Rev. D 103, L031301 (2021).
  51. R. Ruiz, A. Costantini, F. Maltoni, and O. Mattelaer, J. High Energy Phys. 06 (2022) 114.

Outline

Information

Phys. Rev. D 113, 065004– Published 6 March, 2026
  • Received 30 May 2024
  • Accepted 18 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