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  • Letter
  • Open Access

Soft unification of exceptional effective field theories in de Sitter space

Zong-Zhe Du*

  • *Contact author: zongzhe.du@nottingham.ac.uk
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Phys. Rev. D 113, L021701 – Published 7 January, 2026

DOI: https://doi.org/10.1103/s7kf-q4ly

Abstract

We uncover a surprising universal soft behavior (USB) of a de Sitter (dS) S-matrix for all exceptional effective field theories (EFTs) in dS space, stemming from the recently proposed generalized energy conservation condition. At leading order in the soft limit, the dS S matrices for all exceptional EFTs in four spacetime dimensions exhibit no scaling with the soft momentum, i.e., limk→0A(k)∼O(1), a feature we term USB. We specifically show that USB fixes the four-point amplitudes in Dirac-Born-Infeld theory and Special Galileon, and six-point amplitude in SU(N) nonlinear sigma model given the minimal derivative count. We then conjecture USB fixes the interactions of all exceptional EFTs to all orders, thereby providing a unification criterion for them. Our result further underpins the idea that Δ≥4 exceptional theories in dS are characterized by the spectrum and stability requirement alone, resonating with the low energy theory of gravity-general relativity.

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References (33)

  1. Y. Donath and E. Pajer, J. High Energy Phys. 07 (2024) 064.
  2. Z.-Z. Du and D. Stefanyszyn, Phys. Rev. Lett. 135, 201501 (2025).
  3. I. Fujisawa and R. Nakayama, Nucl. Phys. B886, 135 (2014).
  4. T. Basile, X. Bekaert, and N. Boulanger, J. High Energy Phys. 05 (2017) 081.
  5. Z. Sun, Rev. Math. Phys. 37, 2430007 (2025).

  6. Unlike in flat space, the four-point dS amplitude cannot be uniquely fixed by the spectrum, derivative count, and dS invariance, since scale invariance allows different operators to contribute to the same kinematic dependence.

  7. The non–energy-conserving part of the amplitude only has support when energy in the future is greater (lesser) than energy in the past, which implies eternal energy creation(annihilation), leading to potential instability.

  8. G. Goon, K. Hinterbichler, and M. Trodden, J. Cosmol. Astropart. Phys. 07 (2011) 017.
  9. J. Bonifacio, K. Hinterbichler, A. Joyce, and R. A. Rosen, J. High Energy Phys. 02 (2019) 178.
  10. J. Bonifacio, K. Hinterbichler, A. Joyce, and D. Roest, J. High Energy Phys. 04 (2022) 128.
  11. S. L. Adler, Phys. Rev. 139, B1638 (1965).
  12. C. Cheung, K. Kampf, J. Novotny, and J. Trnka, Phys. Rev. Lett. 114, 221602 (2015).
  13. C. Cheung, K. Kampf, J. Novotny, C.-H. Shen, and J. Trnka, Phys. Rev. Lett. 116, 041601 (2016).
  14. K. Hinterbichler and A. Joyce, Phys. Rev. D 92, 023503 (2015).
  15. D. Roest, D. Stefanyszyn, and P. Werkman, J. High Energy Phys. 08 (2019) 081.
  16. Z.-Z. Du and D. Stefanyszyn, J. High Energy Phys. 07 (2024) 011.
  17. C. Cheung, K. Kampf, J. Novotny, C.-H. Shen, and J. Trnka, J. High Energy Phys. 02 (2017) 020.
  18. N. Bittermann and A. Joyce, J. High Energy Phys. 03 (2023) 092.
  19. C. Armstrong, A. Lipstein, and J. Mei, J. High Energy Phys. 12 (2022) 064.
  20. See Supplemental Material at http://link.aps.org/supplemental/10.1103/s7kf-q4ly for a detailed derivation of Universal Soft Behavior from Ward-Takahashi identity, which include Refs. [21,22].
  21. L. Hui, A. Joyce, and S. S. C. Wong, J. High Energy Phys. 02 (2019) 060.
  22. Z.-Z. Du, J. High Energy Phys. 03 (2025) 206.
  23. J. A. Cronin, Phys. Rev. 161, 1483 (1967).
  24. S. Weinberg, Phys. Rev. Lett. 18, 188 (1967).
  25. J. E. Paton and H.-M. Chan, Nucl. Phys. B10, 516 (1969).
  26. J. R. Ellis and B. Renner, Nucl. Phys. B21, 205 (1970).
  27. L. Susskind and G. Frye, Phys. Rev. D 1, 1682 (1970).
  28. K. Kampf, J. Novotny, and J. Trnka, J. High Energy Phys. 05 (2013) 032.
  29. C. Bartsch, K. Kampf, J. Novotny, and J. Trnka, Phys. Rev. D 110, 045009 (2024).
  30. K. Kampf, J. Novotny, and J. Trnka, Phys. Rev. D 87, 081701 (2013).
  31. L. Rodina, J. High Energy Phys. 09 (2019) 084.
  32. S. Jazayeri, E. Pajer, and D. Stefanyszyn, J. High Energy Phys. 10 (2021) 065.
  33. S. Deser, Gen. Relativ. Gravit. 1, 9 (1970).

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Phys. Rev. D 113, L021701– Published 7 January, 2026
  • Received 16 September 2025
  • Revised 21 October 2025
  • Accepted 15 December 2025
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