- 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

Computing c- and a-functions from entanglement

Konstantinos Boutivas*, Dimitrios Katsinis†, Georgios Pastras‡,§, and Nikolaos Tetradis║

  • *Contact author: kboutivas@phys.uoa.gr
  • †Contact author: dkatsinis@phys.uoa.gr
  • ‡Also at Laboratory for Manufacturing Systems and Automation, Department of Mechanical Engineering and Aeronautics, University of Patras, Patra 26110, Greece.
  • §Contact author: pastras@lms.mech.upatras.gr
  • ║Contact author: ntetrad@phys.uoa.gr
PDF

Phys. Rev. D 113, 105026 – Published 29 May, 2026

DOI: https://doi.org/10.1103/knz5-m5zq

Abstract

We investigate the connection between entanglement entropy and the notion of irreversibility in the renormalization-group flow in the context of a simple theory for which an explicit calculation from first principles is feasible. The change of the entanglement entropy for a spherical entangling surface as its radius grows from zero to infinity corresponds to the flow from the UV to the IR. Through analytical and numerical means, we compute the entanglement entropy for a free massive scalar theory, making use of the method of correlation functions. We deduce a c-function in 1+1 dimensions and an a-function in 3+1 dimensions. Both functions are monotonic and vary continuously between one and zero, which are the expected values for this theory.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (34)

  1. K. G. Wilson and J. B. Kogut, The renormalization group and the epsilon expansion, Phys. Rep. 12, 75 (1974).
  2. A. B. Zamolodchikov, Irreversibility of the flux of the renormalization group in a 2D field theory, JETP Lett. 43, 730 (1986), http://jetpletters.ru/ps/1413/article_21504.shtml.
  3. J. L. Cardy, Is there a c theorem in four-dimensions?, Phys. Lett. B 215, 749 (1988).
  4. I. Jack and H. Osborn, Analogs for the c theorem for four-dimensional renormalizable field theories, Nucl. Phys. B343, 647 (1990).
  5. Z. Komargodski and A. Schwimmer, On renormalization group flows in four dimensions, J. High Energy Phys. 12 (2011) 099.
  6. T. Hartman and G. Mathys, Averaged null energy and the renormalization group, J. High Energy Phys. 12 (2023) 139.
  7. H. Casini and M. Huerta, A finite entanglement entropy and the c-theorem, Phys. Lett. B 600, 142 (2004).
  8. H. Casini and M. Huerta, Entanglement and alpha entropies for a massive scalar field in two dimensions, J. Stat. Mech. (2005) P12012.
  9. H. Casini and M. Huerta, On the RG running of the entanglement entropy of a circle, Phys. Rev. D 85, 125016 (2012).
  10. H. Casini, E. Testé, and G. Torroba, Markov property of the conformal field theory vacuum and the a theorem, Phys. Rev. Lett. 118, 261602 (2017).
  11. N. Abate and G. Torroba, Irreversibility of quantum field theory in de Sitter: The C, F and A theorems, arXiv:2411.08961.
  12. C. Holzhey, F. Larsen, and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B424, 443 (1994).
  13. V. Korepin, Universality of entropy scaling in one dimensional gapless models, Phys. Rev. Lett. 92, 096402 (2004).
  14. P. Calabrese and J. L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. (2004) P06002.
  15. H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42, 504007 (2009).
  16. S. N. Solodukhin, Entanglement entropy, conformal invariance and extrinsic geometry, Phys. Lett. B 665, 305 (2008).
  17. R. Lohmayer, H. Neuberger, A. Schwimmer, and S. Theisen, Numerical determination of entanglement entropy for a sphere, Phys. Lett. B 685, 222 (2010).
  18. H. Casini and M. Huerta, Entanglement entropy for the n-sphere, Phys. Lett. B 694, 167 (2011).
  19. N. Abate, I. Salazar, and G. Torroba, Entanglement and renormalization group irreversibility of quantum field theory in AdS, arXiv:2603.10117.
  20. R. D. Sorkin, 1983 paper on entanglement entropy: “On the entropy of the vacuum outside a horizon,” in 10th International Conference on General Relativity and Gravitation (1984), Vol. 2, pp. 734–736, arXiv:1402.3589.
  21. L. Bombelli, R. K. Koul, J. Lee, and R. D. Sorkin, A quantum source of entropy for black holes, Phys. Rev. D 34, 373 (1986).
  22. M. Srednicki, Entropy and area, Phys. Rev. Lett. 71, 666 (1993).
  23. I. Peschel, Calculation of reduced density matrices from correlation functions, J. Phys. A 36, L205 (2003).
  24. R. D. Sorkin, Expressing entropy globally in terms of (4D) field-correlations, J. Phys. Conf. Ser. 484, 012004 (2014).
  25. K. Boutivas, D. Katsinis, G. Pastras, and N. Tetradis, Numerical calculation of entanglement entropy in de Sitter space, Phys. Rev. D 111, 065010 (2025).
  26. K. Boutivas, D. Katsinis, G. Pastras, and N. Tetradis, Entanglement entropy as a probe beyond the horizon, Phys. Rev. Lett. 135, 061602 (2025).
  27. K. Boutivas, D. Katsinis, I. Papadimitriou, G. Pastras, and N. Tetradis, Entanglement entropy of a scalar field in anti-de Sitter space, J. High Energy Phys. 11 (2025) 064.
  28. K. Boutivas, D. Katsinis, G. Pastras, N. Prountzos, and N. Tetradis, Entanglement on a sphere, Phys. Rev. D 112, 065020 (2025).
  29. M. Huerta and G. van der Velde, Modular Hamiltonian of the scalar in the semi infinite line: Dimensional reduction for spherically symmetric regions, J. High Energy Phys. 06 (2022) 097.
  30. D. Katsinis and G. Pastras, Entanglement in (1+1)-dimensional free scalar field theory: Tiptoeing between continuum and discrete formulations, Phys. Rev. D 110, 085015 (2024).
  31. N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 1982).
  32. M. P. Hertzberg and F. Wilczek, Some calculable contributions to entanglement entropy, Phys. Rev. Lett. 106, 050404 (2011).
  33. Y. K. Yazdi, Zero modes and entanglement entropy, J. High Energy Phys. 04 (2017) 140.
  34. C. G. Callan, Jr. and F. Wilczek, On geometric entropy, Phys. Lett. B 333, 55 (1994).

Outline

Information

Phys. Rev. D 113, 105026– Published 29 May, 2026
  • Received 25 January 2026
  • Accepted 24 March 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