- 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

ξRϕ2 nonminimal coupling, and the long range gravitational potential for different spin fields from 2-2 scattering amplitudes

Avijit Sen Majumder*, Ayan Kumar Naskar†, and Sourav Bhattacharya‡

  • Relativity and Cosmology Research Centre, Department of Physics, Jadavpur University, Kolkata 700 032, India

  • *Contact author: senmajumderavijit@gmail.com
  • †Contact author: meayannaskar@gmail.com
  • ‡Contact author: sbhatta.physics@jadavpuruniversity.in
PDF

Phys. Rev. D 114, 085021 – Published 21 October, 2026

DOI: https://doi.org/10.1103/zs6q-prh4

Abstract

In this paper, we investigate the long range gravitational effect of curvature-scalar field nonminimal coupling in the form of ξRϕ2, in the perturbative quantum gravity framework. Such coupling is most naturally motivated from the renormalization of a scalar field theory with a quartic self interaction in a curved spacetime background. This coupling results in two scalar-n graviton vertices which contain no explicit momenta of the scalar, qualitatively different from the usual, e.g., κhμνTμν-type minimal matter-graviton vertices. Assuming the dimensionless coupling parameter ξ to be small, we compute the 2-2 scattering Feynman amplitudes between such scalars up to O(G2ξ). From the nonrelativistic limit of these amplitudes, we compute the corresponding long range gravitational potential. There exists no tree level contribution (O(ξG)) here, and hence the one loop O(G2ξ) result is leading. Recently, the effect of a cosmological constant in such nonminimal interaction and the subsequent gravitational potential was computed. In this work, we take the cosmological constant to be vanishing. The resulting potential is found to have r−4 leading behavior. We further extend these results for scalar-massive spin-1 and massive spin-1/2 scattering. Spin and polarization dependence of the two body potential have been explicitly demonstrated. We discuss some possible physical implications of these results.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (88)

  1. R. Utiyama and B. S. DeWitt, Renormalization of a classical gravitational field interacting with quantized matter fields, J. Math. Phys. (N.Y.) 3, 608 (1962).
  2. S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140, B516 (1965).
  3. H. Pagels, Energy-momentum structure form factors of particles, Phys. Rev. 144, 1250 (1966).
  4. H. van Dam and M. J. G. Veltman, Massive and massless Yang-Mills and gravitational fields, Nucl. Phys. B22, 397 (1970).
  5. V. I. Zakharov, Linearized gravitation theory and the graviton mass, JETP Lett. 12, 312 (1970).
  6. Y. Iwasaki, Quantum theory of gravitation vs. classical theory.—fourth-order potential, Prog. Theor. Phys. 46, 1587 (1971).
  7. D. G. Boulware and S. Deser, Can gravitation have a finite range?, Phys. Rev. D 6, 3368 (1972).
  8. G. ’tHooft and M. J. G. Veltman, One-loop divergencies in the theory of gravitation, Ann. Inst. Henri Poincaré, Phys. Theor. A 20, 69 (1974).
  9. S. Deser and P. van Nieuwenhuisen, One-loop divergences of quantized Einstein-Maxwell fields, Phys. Rev. D 10, 401 (1974).
  10. S. Deser and P. van Nieuwenhuisen, Nonrenormalizability of the quantized Dirac-Einstein system, Phys. Rev. D 10, 411 (1974).
  11. S. Deser, H. S. Tsao, and P. van Nieuwenhuizen, One loop divergences of the Einstein Yang-Mills system, Phys. Rev. D 10, 3337 (1974).
  12. K. S. Stelle, Renormalization of higher derivative quantum gravity, Phys. Rev. D 16, 953 (1977).
  13. B. L. Voronov and I. V. Tyutin, On renormalization of R2 gravitation, Sov. J. Nucl. Phys. 39, 998 (1984).
  14. M. H. Goroff and A. Sagnotti, The ultraviolet behavior of Einstein gravity, Nucl. Phys. B266, 709 (1986).
  15. S. D. Odintsov, The Vilkovisky effective action in quantum gravity with SU(5) grand unification theory, Europhys. Lett. 10, 287 (1989).
  16. S. D. Odintsov and I. L. Shapiro, Perturbative analysis of two-dimensional quantum gravity: Renormalization, gauge dependence and finiteness, Int. J. Mod. Phys. D 01, 571 (1992).
  17. A. Eichhorn, Faddeev-Popov ghosts in quantum gravity beyond perturbation theory, Phys. Rev. D 87, 124016 (2013).
  18. P. M. Lavrov and I. L. Shapiro, Gauge invariant renormalizability of quantum gravity, Phys. Rev. D 100, 026018 (2019).
  19. V. Mukhanov and S. Winitzki, Introduction to Quantum Effects in Gravity (Cambridge University Press, Cambridge, England, 2007).
  20. C. Kiefer, Quantum Gravity (Oxford University Press, UK, 2017).
  21. I. L. Buchbinder and I. L. Shapiro, Introduction to Quantum Field Theory with Applications to Quantum Gravity (Oxford University Press, UK, 2021).
  22. K. Hiida and H. Okamura, Gauge transformation and gravitational potentials, Prog. Theor. Phys. 47, 1743 (1972).
  23. B. M. Barker and R. F. O’Connell, Gravitational two-body problem with arbitrary masses, spins, and quadrupole moments, Phys. Rev. D 12, 329 (1975).
  24. J. F. Donoghue, Leading quantum correction to the Newtonian potential, Phys. Rev. Lett. 72, 2996 (1994).
  25. J. F. Donoghue, General relativity as an effective field theory: The leading quantum corrections, Phys. Rev. D 50, 3874 (1994).
  26. S. D. Odintsov, Renormalization group and effective potential in higher derivative quantum gravity with matter, Phys. Lett. B 336, 347 (1994).
  27. E. Elizalde, S. D. Odintsov, and I. L. Shapiro, Asymptotic regimes in quantum gravity at large distances and running Newtonian and cosmological constants, Classical Quantum Gravity 11, 1607 (1994).
  28. I. J. Muzinich and S. Vokos, Long range forces in quantum gravity, Phys. Rev. D 52, 3472 (1995).
  29. H. W. Hamber and S. Liu, On the quantum corrections to the Newtonian potential, Phys. Lett. B 357, 51 (1995).
  30. G. Modanese, Potential energy in quantum gravity, Nucl. Phys. B434, 697 (1995).
  31. A. A. Akhundov, S. Bellucci, and A. Shiekh, Gravitational interaction to one loop in effective quantum gravity, Phys. Lett. B 395, 16 (1997).
  32. J. F. Donoghue and T. Torma, Power counting of loop diagrams in general relativity, Phys. Rev. D 54, 4963 (1996).
  33. N. E. J. Bjerrum-Bohr, Leading quantum gravitational corrections to scalar QED, Phys. Rev. D 66, 084023 (2002).
  34. N. E. J. Bjerrum-Bohr, J. F. Donoghue, and B. R. Holstein, Quantum corrections to the Schwarzschild and Kerr metrics, Phys. Rev. D 68, 084005 (2003).
  35. N. E. J. Bjerrum Bohr, J. F. Donoghue, and B. R. Holstein, Quantum gravitational corrections to the nonrelativistic scattering potential of two masses, Phys. Rev. D 67, 084033 (2003).
  36. Z. Bern, Perturbative quantum gravity and its relation to gauge theory, Living Rev. Relativity 5, 5 (2002).
  37. A. Akhundov and A. Shiekh, A review of leading quantum gravitational corrections to Newtonian gravity, Electron. J. Theor. Phys. 17, 1 (2008).
  38. B. R. Holstein and A. Ross, Spin effects in long range gravitational scattering, arXiv:0802.0716.
  39. D. J. Toms, Quantum gravitational contributions to quantum electrodynamics, Nature (London) 468, 56 (2010).
  40. N. E. J. Bjerrum-Bohr, J. F. Donoghue, B. K. El-Menoufi, B. R. Holstein, L. Planté, and P. Vanhove, The equivalence principle in a quantum world, Int. J. Mod. Phys. D 24, 1544013 (2015).
  41. P. C. Malta and L. P. R. Ospedal and K. P. B. Veiga, and J. A. Helayel-Neto, Comparative aspects of spin-dependent interaction potentials for spin-1/2 and spin-1 matter fields, Adv. High Energy Phys. 2016, 2531436 (2016).
  42. M. B. Fröb, Quantum gravitational corrections for spinning particles, J. High Energy Phys. 10 (2016) 051.
  43. S. C. Ulhoa, A. F. Santos, and F. C. Khanna, Scattering of fermions by gravitons, Gen. Relativ. Gravit. 49, 54 (2017).
  44. T. Olyaei and A. Aziziy, Weak gravitational interaction of fermions: Quantum viewpoint, Mod. Phys. Lett. A 33, 1850218 (2018).
  45. G. P. de Brito, M. G. Campos, L. P. R. Ospedal, and K. P. B. Veiga, Quantum corrected gravitational potential beyond monopole-monopole interactions, Phys. Rev. D 102, 084015 (2020).
  46. M. B. Fröb, Graviton corrections to the Newtonian potential using invariant observables, J. High Energy Phys. 01 (2022) 180.
  47. A. S. Majumder and S. Bhattacharya, Scattering of massive spin-2 field via graviton exchanges with different spin fields and the gravitational potential, Phys. Rev. D 113, 125004 (2026).
  48. J. F. Donoghue, M. M. Ivanov, and A. Shkerin, EPFL lectures on general relativity as a quantum field theory, arXiv:1702.00319.
  49. N. E. J. Bjerrum-Bohr, J. F. Donoghue, B. R. Holstein, L. Planté, and P. Vanhove, Bending of light in quantum gravity, Phys. Rev. Lett. 114, 061301 (2015).
  50. Dong Bai and Yue Huang, More on the bending of light in quantum gravity, Phys. Rev. D 95, 064045 (2017).
  51. F. Bastianelli, F. Comberiati, and L. de la Cruz, Light bending from eikonal in worldline quantum field theory, J. High Energy Phys. 02 (2022) 209.
  52. E. Elizalde, S. D. Odintsov, and I. L. Shapiro, Asymptotic regimes in quantum gravity at large distances and running Newtonian and cosmological constants, Classical Quantum Gravity 11, 1607 (1994).
  53. D. J. Toms, Cosmological constant and quantum gravitational corrections to the running fine structure constant, Phys. Rev. Lett. 101, 131301 (2008).
  54. D. J. Toms, Low energy quantum gravity, the cosmological constant and gauge coupling constants, Int. J. Mod. Phys. D 17, 2447 (2009).
  55. D. J. Toms, Quantum gravity, gauge coupling constants, and the cosmological constant, Phys. Rev. D 80, 064040 (2009).
  56. D. J. Toms, Quadratic divergences and quantum gravitational contributions to gauge coupling constants, Phys. Rev. D 84, 084016 (2011).
  57. A. S. Majumder and S. Bhattacharya, ξRϕ2 coupling, cosmological constant and quantum gravitational corrections to Newton’s potential, Phys. Lett. B 873, 140147 (2026).
  58. J. F. Donoghue, Introduction to the effective field theory description of gravity, arXiv:gr-qc/9512024.
  59. C. P. Burgess, Quantum gravity in everyday life: General relativity as an effective field theory, Living Rev. Relativity 7, 5 (2004).
  60. W. D. Goldberger and I. Z. Rothstein, An effective field theory of gravity for extended objects, Phys. Rev. D 73, 104029 (2006).
  61. S. Foffa, P. Mastrolia, R. Sturani, and C. Sturm, Effective field theory approach to the gravitational two-body dynamics, at fourth post-Newtonian order and quintic in the Newton constant, Phys. Rev. D 95, 104009 (2017).
  62. M. Levi, Effective field theories of post-Newtonian gravity: A comprehensive review, Rep. Prog. Phys. 83, 075901 (2020).
  63. C. Cheung, I. Z. Rothstein, and M. P. Solon, From scattering amplitudes to classical potentials in the post-Minkowskian expansion, Phys. Rev. Lett. 121, 251101 (2018).
  64. Z. Bern, A. Luna, R. Roiban, C. H. Shen, and M. Zeng, Spinning black hole binary dynamics, scattering amplitudes, and effective field theory, Phys. Rev. D 104, 065014 (2021).
  65. M. M. Ivanov and Z. Zhou, Vanishing of black hole tidal Love numbers from scattering amplitudes, Phys. Rev. Lett. 130, 091403 (2023).
  66. G. L. Almeida, Binary dynamics to second post-Newtonian order in scalar-tensor and Einstein-scalar-Gauss-Bonnet gravity from effective field theory, Phys. Rev. D 109, 8 (2024).
  67. G. L. Almeida and S. Y. Zhou, Post-Newtonian dynamics of spinning black hole binaries in Einstein-scalar-Gauss-Bonnet gravity, Phys. Rev. D 110, 124016 (2024).
  68. D. Trestini, Gravitational waves from quasielliptic compact binaries in scalar-tensor theory to one-and-a-half post-Newtonian order, Classical Quantum Gravity 42, 155016 (2025).
  69. I. L. Buchbinder, S. D. Odintsov, and O. A. Fonarev, The behavior of the effective coupling constants for E(6) GUT in curved space-time, Mod. Phys. Lett. A 04, 2713 (1989).
  70. T. Muta and S. D. Odintsov, Model dependence of the nonminimal scalar graviton effective coupling constant in curved space-time, Mod. Phys. Lett. A 06, 3641 (1991).
  71. A. A. Bytsenko, S. D. Odintsov, and S. Zerbini, The large distance limit of the gravitational effective action in hyperbolic backgrounds, Classical Quantum Gravity 12, 1 (1995).
  72. E. Elizalde, S. D. Odintsov, and A. Romeo, Improved effective potential in curved space-time and quantum matter, higher derivative gravity theory, Phys. Rev. D 51, 1680 (1995).
  73. D. Seery, One-loop corrections to a scalar field during inflation, J. Cosmol. Astropart. Phys. 11 (2007) 025.
  74. R. P. Woodard, Perturbative quantum gravity comes of age, Int. J. Mod. Phys. D 23, 1430020 (2014).
  75. K. Y. Oda and M. Yamada, Non-minimal coupling in Higgs–Yukawa model with asymptotically safe gravity, Classical Quantum Gravity 33, 125011 (2016).
  76. I. G. Moss, Covariant one-loop quantum gravity and Higgs inflation, arXiv:1409.2108.
  77. I. L. Shapiro, P. Morais Teixeira, and A. Wipf, On the functional renormalization group for the scalar field on curved background with non-minimal interaction, Eur. Phys. J. C 75, 262 (2015).
  78. I. D. Saltas, Higgs inflation and quantum gravity: An exact renormalization group approach, J. Cosmol. Astropart. Phys. 02 (2016) 048.
  79. I. L. Buchbinder, S. D. Odintsov, and I. L. Shapiro, Effective Action in Quantum Gravity (Routledge, New York, 2017), ISBN [Amazon][WorldCat], [Amazon][WorldCat], [Amazon][WorldCat].
  80. A. Arbuzov, B. Latosh, and A. Nikitenko, Effective potential of scalar-tensor gravity with quartic self-interaction of scalar field, Classical Quantum Gravity 39, 055003 (2022).
  81. L. E. Parker and D. Toms, Quantum Field Theory in Curved Spacetime: Quantized Field and Gravity (Cambridge University Press, Cambridge, England, 2009), ISBN [Amazon][WorldCat], [Amazon][WorldCat], [Amazon][WorldCat].
  82. N. M. Bocharova, K. A. Bronnikov, and V. N. Melnikov, On an exact solution of the system of Einstein equations and a massless scalar field, Vestn. Mosk. Univ., Ser. 3: Fiz., Astron. 6, 706 (1970).
  83. J. D. Bekenstein, Exact solutions of Einstein conformal scalar equations, Ann. Phys. (N.Y.) 82, 535 (1974).
  84. C. Martinez, R. Troncoso, and J. Zanelli, De Sitter black hole with a conformally coupled scalar field in four-dimensions, Phys. Rev. D 67, 024008 (2003).
  85. S. Bhattacharya and H. Maeda, Can a black hole with conformal scalar hair rotate?, Phys. Rev. D 89, 087501 (2014).
  86. M. Bravo-Gaete, F. F. Santos, J. A. Herrera-Mendoza, and D. F. Higuita-Borja, A new rotating axionic AdS4 black hole dressed with a scalar field, Classical Quantum Gravity 43, 075011 (2026).
  87. S. Coleman et al., Lectures of Sidney Coleman on Quantum Field Theory (World Scientific, New Jersey, 2018).
  88. T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, Modified gravity and cosmology, Phys. Rep. 513, 1 (2012).

Outline

Information

Phys. Rev. D 114, 085021– Published 21 October, 2026
  • Received 29 June 2026
  • Accepted 1 October 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