- 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

Yukawa potential of a nonhomogeneous sphere with new limits on an ultralight boson

Pierre Fayet*

  • *Contact author: pierre.fayet@phys.ens.fr
PDF

Phys. Rev. D 112, 095018 – Published 14 November, 2025

DOI: https://doi.org/10.1103/17wc-xxjm

Abstract

Extremely weak long-range forces may lead to apparent violations of the equivalence principle. The final MICROSCOPE  result, leading at 95% CL to |δ|<4.5×10−15 or 6.5×10−15 for a positive or negative Eötvös parameter δ, requires taking into account the spin of the mediator, and the sign of Δ(Q/Ar)Ti−Pt (Q denoting the new charge involved). A coupling to B−L or B should verify |gB−L|<1.1×10−25 or |gB|<8×10−25, for a spin-1 mediator of mass m<10−14  eV/c2, with slightly different limits of 1.3×10−25 or 6.6×10−25 in the spin-0 case. The limits increase with m, in a way which depends on the density distribution within the Earth. This involves an hyperbolic form factor, expressed through a bilateral Laplace transform as Φ(x=mR)=⟨sinhmr/mr⟩, related by analytic continuation to the Earth form factor Φ(ix)=⟨sinmr/mr⟩. It may be expressed as Φ(x)=3x2(coshx−sinhxx)×ρ¯(x)/ρ0, where ρ¯(x) is an effective density, decreasing from the average ρ0 at m=0 down to the density at the periphery. We give general integral or multishell expressions of Φ(x), evaluating it, and ρ¯(x), in a simplified 5-shell model. Φ(x) may be expanded as ∑x2n(2n+1)!⟨r2n⟩R2n≃1+.0827x2+.00271x4+4.78×10−5x6+5.26×10−7x8+⋯, absolutely convergent for all x and potentially useful up to x≈5. The coupling limits increase at large x like mR emz/2/1+mr (z=r−R being the satellite altitude), getting multiplied by ≃ 1.9, 34, or 1.2×109, for m=10−13,10−12 or 10−11  eV/c2, respectively.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (43)

  1. L. Eötvös, D. Pekár, and E. Fekete, Beiträge zum Gesetz der Proportionalität von Trägheit and Gravität, Ann. Phys. (N.Y.) 68, 11 (1922).
  2. E. Adelberger, C. W. Stubbs, W. F. Rogers, F. J. Raab, B. R. Heckel, J. H. Gundlach, H. E. Swanson, and R. Watanabe, New constraints on composition-dependent interactions weaker than gravity, Phys. Rev. Lett. 59, 849 (1987).
  3. E. Adelberger et al., Testing the equivalence principle in the field of the Earth: Particle physics at masses below 1  μeV?, Phys. Rev. D 42, 3267 (1990).
  4. T. Wagner, S. Schlamminger, J. Gundlach, and E. Adelberger, Torsion-balance tests of the weak equivalence principle, Classical Quantum Gravity 29, 184002 (2012).
  5. P. Touboul et al., MICROSCOPE mission: First results of a space test of the equivalence principle, Phys. Rev. Lett. 119, 231101 (2017); Space test of the equivalence principle: First results of the MICROSCOPE mission, Classical Quantum Gravity 36, 225006 (2019).
  6. P. Touboul et al., MICROSCOPE mission: Final results of the test of the equivalence principle, Phys. Rev. Lett. 129, 121102 (2022); Result of the MICROSCOPE weak equivalence principle test, Classical Quantum Gravity 39, 204009 (2022).
  7. P. Fayet, On the search for a new spin-1 boson, Nucl. Phys. B187, 184 (1981); The fifth force charge as a linear combination of B, L (or B−L) and electric charges, Phys. Lett. B 227, 127 (1989); Extra U(1)’s and new forces, Nucl. Phys. B347, 743 (1990).
  8. P. Fayet, MICROSCOPE limits for new long-range forces and implications for unified theories, Phys. Rev. D 97, 055039 (2018); MICROSCOPE limits on the strength of a new force, with comparisons to gravity and electromagnetism, 99, 055043 (2019).
  9. P. Fayet, The fifth interaction in grand unified theories: A new force acting mostly on neutrons and particle spins, Phys. Lett. B 172, 363 (1986).
  10. P. Fayet, New interactions and the standard models, Classical Quantum Gravity 13, A19 (1996).
  11. R. Abbott et al. (LIGO, Virgo, and KAGRA Collaborations), Constraints on dark photon dark matter using data from LIGO’s and Virgo’s third observing run, Phys. Rev. D 105, 063030 (2022).
  12. E. Shaw, M.  P. Ross, C.  A. Hagedorn, E.  G. Adelberger, and J.  H. Gundlach, Torsion-balance search for ultralow-mass bosonic dark matter, Phys. Rev. D 105, 042007 (2022).
  13. X. Xue et al., High-precision search for dark photon dark matter with the Parkes Pulsar Timing Array, Phys. Rev. Res. 4, L012022 (2022).
  14. A. Afzalet et al. (NANOGrav Collaboration), The NANOGrav 15 yr data set: Search for signals from new physics, Astrophys. J. Lett. 951, 1 (2023).
  15. J. Yu, Y. Yao, Y. Tang, and Y. Wu, Sensitivity of space-based gravitational-wave interferometers to ultralight bosonic fields and dark matter, Phys. Rev. D 108, 083007 (2023).
  16. J. Frerick, J. Jaeckel, F. Kahlhoefer, and K. Schmidt-Hoberg, Riding the dark matter wave: Novel limits on general dark photons from LISA Pathfinder, Phys. Lett. B 848, 138328 (2024).
  17. A. Allocca et al., Thermal noise-limited beam balance as prototype of the Archimedes vacuum weight experiment and B−L dark photon search, Eur. Phys. J. Plus 139, 158 (2024).
  18. L. Sun, B. Slagmolen, and J. Qin, Search for ultralight vector dark matter with a torsion pendulum dual oscillator, Phys. Rev. D 111, 063064 (2025).
  19. D. Amaral, M. Jain, M. Amin, and C. Tunnell, Vector wave dark matter and terrestrial quantum sensors, J. Cosmol. Astropart. Phys. 06 (2024) 050.
  20. A. Abac et al. (LIGO, Virgo, and KAGRA Collaborations), Ultralight vector dark matter search using data from the KAGRA O3 GK run, Phys. Rev. D 110, 042001 (2024).
  21. Y. Okuma, K. Izumi, K. Komori, and M. Ando, Cross-correlated force measurement for thermal noise reduction in torsion pendulum, Phys. Rev. D 111, 082006 (2025).
  22. H. An, T. Li, J. Shu, X. Wang, X. Xue, and Y. Zhao, Dark photon dark matter and low-frequency gravitational wave detection with Gaia-like astrometry, Astrophys. J. 976, 247 (2024).
  23. D. Amaral, D. Uitenbroek, T. Oosterkamp, and C. Tunnell, First search for ultralight dark matter using a magnetically levitated particle, Phys. Rev. Lett. 134, 251001 (2025).
  24. T. Damour and A. Polyakov, The string dilaton and a least coupling principle, Nucl. Phys. B423, 532 (1994).
  25. J. Khoury and A. Weltman, Chameleon fields: Awaiting surprises for tests of gravity in space, Phys. Rev. Lett. 93, 171104 (2004).
  26. T. Damour and J. Donoghue, Equivalence principle violations and couplings of a light dilaton, Phys. Rev. D 82, 084033 (2010).

  27. For a symmetric distribution the above 29.24% would be replaced by the 50% probability for an unconstrained δ to be positive, Eq. (19) simplifying into x=2 erf−1(cl), providing the usual 90% and 95% CL limits at xσ≃1.645 σ and 1.960 σ.

  28. Strictly speaking for a δ of a given sign the limits on |εL| are larger than for |εB−L| by about 2%, a difference that we shall usually consider as negligible.

  29. These 95% CL limits of 1.1 and 1.3×10−25 in the spin-1 and spin-0 cases for a B−L coupling, rounded from 1.07 and 1.28×10−25, differ by a factor ≃6.5/4.5≃1.20. And similarly for the 6.4 and 7.7×10−25 limits for a coupling to B in the spin-0 and spin-1 cases, which also differ by the same factor.

  30. C. Bouwkamp, A new method for computing the energy of interaction between two spheres under a general law of force, Physica (Amsterdam) 13, 501 (1947).
  31. I. Sneddon and C. Thornhill, A property of the Yukawa potential, Math. Proc. Cambridge Philos. Soc. 45, 318 (1949).
  32. E. Fischbach, D. Sudarsky, A. Szafer, C. Talmadge, and S. H. Aronson, Reanalysis of the Eötvös experiment, Phys. Rev. Lett. 56, 3 (1986).
  33. E. Fischbach and C. Talmadge, The Search for Non-Newtonian Gravity (Springer, New York, 1999).
  34. E. Adelberger, B. Heckel, and A. Nelson, Tests of the gravitational inverse square law, Annu. Rev. Nucl. Part. Sci. 53, 77 (2003).

  35. The continuity of V and dV/dr at r=R is expressed as Vin(R)=ρ∘m2x[x−(x+1)e−xsinhx]=ρ∘2m2x[x−1+(x+1)e−2x]=ρ∘m2x(xcoshx−sinhx)e−x=Vout(R),−dVindr(R)=ρ∘m(x+1)e−x(xcoshx−sinhx)x2=−dVoutdr(R).

  36. For large λ≫R i.e. small x=R/λ, ϕ(x)→1 and Vout(r) tends to be the same Yukawa potential as for a pointlike sphere. As sinhmr/mr≃1+x2r2/6R2, we also recover in this massless limit the same inside potential Vin(r)=Q4πR(32−12r2R2) as for a uniformly charged sphere.

  37. M. Pirenne, The Diffraction of X-Rays and Electrons by Free Molecules (Cambridge University Press, London, 1946), p. 12.
  38. A. Dziewonski and D. Anderson, Preliminary reference Earth model, Phys. Earth Planetary Interiors 25, 297 (1981).
  39. NASA Earth fact sheet, https://atoc.colorado.edu/~fasullo/1060/resources/earthfact.html (2024).
  40. C. Denis et al., Hydrostatic flattening, core structure, and translational mode of the inner core, Phys. Earth Planetary Interiors 99, 195 (1997).
  41. B. Kennett, On the density distribution within the Earth, Geophys. J. Int. 132, 374 (1998).
  42. P. Fayet, A new long range force?, Phys. Lett. B 171, 261 (1986),

  43. With m=10−12  eV/c2 corresponding to λ≃197.327  km and x≃32.287, ϕ(x)≃1.466×1011, Φ(x)≃9.13×1010, and ρ¯(x)/ρ0≃.623 as in Tables 4 and 5, i.e., an effective density ρ¯(x)≃3.43  g/cm3 and ρ0/ρ¯(x)≃1.267. With (1+mr)e−mr≃9.60×10−15, the new force gets multiplied by F(x)≃8.77×10−4, and the coupling limits increase by the factor G(x)=F(x)−1/2≃33.8 as in Table 5, also in agreement with (97).

Outline

Information

Phys. Rev. D 112, 095018– Published 14 November, 2025
  • Received 4 July 2025
  • Accepted 5 September 2025
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