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arxiv: 2511.16103 · v2 · submitted 2025-11-20 · ✦ hep-th · gr-qc

Scattering of massive spin-2 field via graviton exchanges with different spin fields and the long range gravitational potential

Avijit Sen Majumder , Sourav Bhattacharya This is my paper

Pith reviewed 2026-05-17 21:09 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords gravitational potentialspin-2 fieldgraviton exchangescattering amplitudenon-relativistic limitFierz-Pauli fieldhigher spin
0
0 comments X

The pith

Graviton exchanges produce the Newtonian potential plus spin-dependent corrections for a massive spin-2 test field.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper computes tree-level scattering amplitudes in which a massive spin-2 Fierz-Pauli field exchanges a single graviton with massive scalars, vectors, and fermions. In the non-relativistic limit these amplitudes directly supply the two-body gravitational potential, including the leading 1/r term and the subleading terms that depend on the spins or polarizations of the particles. A separate calculation at next-to-leading order for the spin-2–scalar system isolates the spin-independent, spherically symmetric piece at order G squared. The massive spin-2 field is treated throughout as a test quantum field minimally coupled to ordinary general relativity, not as the graviton itself. The resulting potentials therefore test how higher-spin matter responds to gravity at the level of scattering.

Core claim

The two-body gravitational potential between a massive spin-2 Fierz-Pauli field and other massive fields is obtained by taking the non-relativistic limit of the tree-level graviton-exchange scattering amplitudes. At order G the potential contains the standard Newtonian attraction together with polarization-dependent corrections; at order G squared the spin-independent, spherically symmetric term is recovered for scattering with a scalar.

What carries the argument

Non-relativistic reduction of the single-graviton-exchange scattering amplitudes in the minimally coupled theory of a massive Fierz-Pauli spin-2 field with general relativity.

If this is right

  • The potential between spin-2 and spin-1/2 particles contains polarization-dependent forces at order G.
  • The same method yields post-Newtonian corrections once higher-order graviton exchanges are included.
  • The framework extends in principle to any massive higher-spin test field coupled to gravity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same scattering technique could be applied to check consistency of gravitational potentials in theories containing multiple massive spin-2 fields.
  • Precision measurements of spin-dependent forces between polarized objects, if ever feasible, would provide a direct test of the subleading terms.

Load-bearing premise

The massive spin-2 field couples to gravity only through the standard minimal prescription, so that the non-relativistic limit of the scattering amplitudes directly yields the potential without extra corrections from the field’s mass or self-interactions.

What would settle it

An explicit mismatch between the non-relativistic reduction of the computed O(G) amplitude and the expected Newtonian term plus spin-dependent pieces would falsify the extraction procedure.

Figures

Figures reproduced from arXiv: 2511.16103 by Avijit Sen Majumder, Sourav Bhattacharya.

Figure 1
Figure 1. Figure 1: The tree level diagram of massive spin-0-spin-0 scattering. The straight lines stand for the scalar field, whereas the wavy line represents the graviton. q = k1 − k ′ 1 = k ′ 2 − k2 is the transfer momentum. V (1) µν spin-0 (k, k′ , m) = − iκ 2 h k µ k ′ν + k ′µ k ν − η µν k · k ′ + m2  i , (15) for the two scalar one graviton vertex, e.g. [26], the Feynman amplitude for this process is given by iM(1)(κ 2… view at source ↗
Figure 2
Figure 2. Figure 2: The tree level scattering of massive spin-2-spin-0 fields. The double lines represent the massive spin-2 field. Let us now compute the lowest order scattering amplitude between a massive scalar and a massive spin-2 field, [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The ladder (left) and the cross-ladder diagram for massive spin-2-spin-0 fields scattering. As earlier, double and single straight lines respectively stand for the massive spin-2 and spin-0 fields, whereas a wavy line represents the graviton. q = k1 − k ′ 1 = k ′ 2 − k2 is the transfer momentum. where PM is given by Eq. 12, and V¯ (1) spin-2 µν,eff. contains no polarisation tensor. We have, after some alge… view at source ↗
Figure 4
Figure 4. Figure 4: The triangle diagrams for massive spin-2-spin-0 fields scattering. and, iMtriangle-2 = Z d 4 l (2π) 4 V µναβ (2)spin-2(k1, k′ 1 , M)V σρ (1)spin-0(k2, k2 − l, m)V γδ (1)spin-0(k2 − l, k′ 2 , m) × −iPµνσρ l 2 −iPαβγδ (l + q) 2 −i (l − k2) 2 + m2 , (50) where V µναβ (2)spin-2(k1, k′ 1 , M) is the effective two massive spin-2-two graviton vertex containing two polarisations ϵµν(k1) and ϵ ⋆ αβ(k ′ 1 ), and can… view at source ↗
Figure 5
Figure 5. Figure 5: The double seagull diagram for massive spin-2-spin-0 fields scattering. iMdouble-seagull = 1 2! Z d 4 l (2π) 4 V αβγδ (2)spin-2(k1, k′ 1 , M)V σρµν (2)spin-0(k2, k′ 2 , m) −iPαβµν (l + q) 2 −iPγδσρ l 2 , (54) where as earlier V αβγδ (2)spin-2(k1, k′ 1 , M) contains contractions with two polarisations of the massive spin-2 field. The above expression yields, Mdouble-seagull = − 1 8 G 2m2M2 ln q 2 ϵ µνϵ ∗ µν… view at source ↗
Figure 6
Figure 6. Figure 6: The first vertex correction diagram for massive spin-2-spin-0 fields scattering. which yields, Mvertex-correction-1 = " 46π 2G2m3M2 q + 4π 2G2m2M3 q + 46G 2m2M2 ln q 2 # ϵ µν(k1)ϵ ∗ µν(k ′ 1 ) + polarisation − dependent subleading terms. (59) The corresponding correction to the Newton potential reads, Vvertex-correction-1(r) =" − 23G2m2M 4r 2 − G2mM2 2r 2 + 23G2mM 4πr3 # (⃗ϵ · ⃗ϵ ′ ) 2 + polarisation − dep… view at source ↗
Figure 7
Figure 7. Figure 7: The second vertex correction diagram for massive spin-2-spin-0 fields scattering. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The third vertex correction diagram for massive spin-2-spin-0 fields scattering. iMvertex-correction-3 = 1 2! Z d 4 l (2π) 4 V (1)spin-2, eff. αβ (k1, k′ 1 , M) −iP αβ γδ (l + q) 2 V (3)γδ µνρσ (l, −q) −iP ρσϵϕ q 2 −iP µνλψ l 2 × V (1)spin-0 λψϵϕ (k2, k′ 2 , m), (64) which gives, Mvertex-correction-3 = 80G 2m2M2 ln q 2 ϵ µν(k ′ 1 )ϵ ∗ µν(k ′ 1 ) + polarisation − dependent subleading terms (65) The correspo… view at source ↗
Figure 9
Figure 9. Figure 9: The fourth vertex correction diagram for massive spin-2-spin-0 fields scattering. 4.7 The vertex correction diagram-4 The fourth and the last vertex correction diagram is given by [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The vacuum polarisation plus ghost diagrams for massive spin-2-spin-0 fields scattering. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Graviton self energy from the vacuum polarisation diagram along with the ghost contribution. iMvac-pol = V (1)spin-2, eff. ρσ (k1, k′ 1 , M) −iP ρσλξ q 2 Πλξµν(q) −iP µνγδ q 2 V (1)spin -0 γδ (k2, k′ 2 , m), (71) which equals after substituting Eq. 70, MGhost = − 172 15 G 2m2M2 ln q 2 ϵ µν(k1)ϵ ∗ µν(k ′ 1 ) + polarisation − dependent subleading terms. (72) The corresponding potential reads, Vvac-pol(r) = … view at source ↗
read the original abstract

In this work, we compute the graviton mediated scattering amplitude of a massive spin-2 Fierz-Pauli field with various other massive spin fields, and in the non-relativistic limit, find out the corresponding two-body gravitational potentials. The massive spin-2 field does not represent gravity here. The theory of gravity is taken to be the usual massless general relativity, and the massive spin-2 field is taken as a test quantum field coupled to gravity via the standard minimal prescription. We first compute the tree level 2-2 scattering of a massive spin-2 field with massive scalar, spin-1, and spin-1/2 fields with one graviton exchanges. Leading Newton potential, as well as the subleading spin or polarisation dependent terms at ${\cal O}(G)$ have been computed. We also consider the next to the leading order (${\cal O}(G^2)$) scattering of the massive spin-2 field with a massive scalar, and demonstrate the spin independent, spherically symmetric leading part of the two body gravitational potential. The present paper can be considered as an attempt to compute the gravitational potential in the context of a higher spin field theory.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The paper computes tree-level 2-to-2 scattering amplitudes of a massive Fierz-Pauli spin-2 field with massive scalar, spin-1, and spin-1/2 fields mediated by single graviton exchange in massless GR. From these amplitudes the authors extract the non-relativistic two-body gravitational potentials, reporting the leading Newtonian term together with sub-leading spin- or polarization-dependent corrections at O(G). They additionally evaluate the O(G²) amplitude for the spin-2–scalar system and isolate the spin-independent, spherically symmetric piece of the resulting potential.

Significance. If the calculations are correct, the work supplies explicit analytic expressions for gravitational potentials felt by a massive spin-2 test particle, extending the classic scalar, vector and fermion cases. The O(G²) result for the scalar channel is a modest but concrete step toward higher-order potentials in the presence of higher-spin matter. The manuscript is a straightforward application of standard Feynman rules and non-relativistic reduction; its main value lies in the concrete expressions rather than in conceptual novelty.

major comments (1)
  1. [non-relativistic reduction] The non-relativistic reduction is performed directly on the tree-level amplitudes. For a rank-2 tensor field the propagator and graviton vertices contain multiple independent tensor structures; an incomplete expansion in |p|/m could leave residual mass-dependent terms that do not appear for lower-spin fields. Explicit verification that the leading spin-independent potential at both O(G) and O(G²) is free of such corrections would strengthen the central claim (see the paragraph following Eq. (X) in the O(G) section and the O(G²) discussion).
minor comments (3)
  1. [Abstract] The abstract states that the massive spin-2 field is a test field coupled via the minimal prescription; this should be repeated once in the introduction for clarity.
  2. [amplitude computation] Several intermediate steps in the amplitude calculation (contraction of polarization tensors, trace terms, etc.) are omitted. Adding a short appendix with the key contractions would improve reproducibility.
  3. [Introduction] The paper would benefit from a brief comparison with existing literature on gravitational potentials for higher-spin fields (e.g., works on massive spin-2 in curved backgrounds).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comment on the non-relativistic reduction. The recommendation for minor revision is appreciated, and we address the point raised below.

read point-by-point responses

  1. Referee: [non-relativistic reduction] The non-relativistic reduction is performed directly on the tree-level amplitudes. For a rank-2 tensor field the propagator and graviton vertices contain multiple independent tensor structures; an incomplete expansion in |p|/m could leave residual mass-dependent terms that do not appear for lower-spin fields. Explicit verification that the leading spin-independent potential at both O(G) and O(G²) is free of such corrections would strengthen the central claim (see the paragraph following Eq. (X) in the O(G) section and the O(G²) discussion).

    Authors: We thank the referee for highlighting this subtlety specific to the rank-2 field. In deriving the potentials we performed a systematic expansion of the full tree-level amplitudes (including all tensor structures in the massive spin-2 propagator and the graviton vertices) in powers of |p|/m, keeping only the terms that survive in the non-relativistic limit. After projecting onto the appropriate polarization tensors and taking the appropriate limits, the leading spin-independent pieces at both O(G) and O(G²) reduce exactly to the Newtonian form -G m1 m2/r with no residual mass-dependent corrections. To make this verification explicit we will add a short clarifying paragraph immediately after the relevant equations in the revised manuscript, together with a brief outline of the expansion steps that eliminate potential artifacts. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper computes explicit tree-level graviton-exchange amplitudes for a massive Fierz-Pauli spin-2 test field coupled minimally to standard GR, then extracts the non-relativistic two-body potentials at O(G) and O(G^2) directly from those amplitudes. No parameters are fitted to data and then relabeled as predictions, no self-definitional loops appear in the propagator or vertex expansions, and no load-bearing uniqueness theorem or ansatz is imported via self-citation. The central results follow from standard Feynman rules and the usual |p|/m expansion without reducing to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Calculation rests on standard QFT and GR assumptions without new free parameters or postulated entities.

axioms (2)
  • domain assumption Minimal coupling of the massive spin-2 field to the graviton via the standard prescription in flat-space GR
    Invoked to define the interaction vertices for the test field.
  • domain assumption Validity of the non-relativistic limit on the scattering amplitudes to extract two-body potentials
    Used to obtain the leading Newton term and subleading corrections.

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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    hep-th 2026-04 unverdicted novelty 5.0

    The ξ R φ² non-minimal coupling produces a leading r^{-4} long-range gravitational potential from one-loop 2-2 scattering amplitudes in perturbative quantum gravity with vanishing cosmological constant, with explicit ...

Reference graph

Works this paper leans on

65 extracted references · 65 canonical work pages · cited by 1 Pith paper · 19 internal anchors

  1. [1]

    Utiyama and B

    R. Utiyama and B. S. DeWitt,Renormalization of a classical gravitational field interacting with quantized matter fields, J. Math. Phys.3, 608 (1962)

    work page 1962

  2. [2]

    Weinberg,Infrared photons and gravitons, Phys

    S. Weinberg,Infrared photons and gravitons, Phys. Rev.140, B516 (1965)

    work page 1965

  3. [3]

    D. G. Boulware and S. Deser,Can gravitation have a finite range?, Phys. Rev. D6, 3368–3382 (1972)

    work page 1972

  4. [4]

    ’tHooft and M

    G. ’tHooft and M. J. G. Veltman,One-loop divergencies in the theory of gravitation, Ann. Inst. H. Poincar´ e, Phys. Theor. A20, 69–94 (1974)

    work page 1974

  5. [5]

    Deser and P

    S. Deser and P. van Nieuwenhuisen,One-loop divergences of quantized Einstein-Maxwell fields, Phys. Rev. D10, 401 (1974)

    work page 1974

  6. [6]

    Deser and P

    S. Deser and P. van Nieuwenhuisen,Nonrenormalizability of the quantized Dirac-Einstein system, Phys. Rev. D10, 411 (1974)

    work page 1974

  7. [7]

    Deser, H

    S. Deser, H. S. Tsao and P. van Nieuwenhuizen,One loop divergences of the Einstein Yang-Mills system, Phys. Rev. D10, 3337 (1974)

    work page 1974

  8. [8]

    K. S. Stelle,Renormalization of higher derivative quantum gravity, Phys. Rev. D16, 953 (1977)

    work page 1977

  9. [9]

    B. L. Voronov and I. V. Tyutin,On renormalization ofR 2 gravitation, Sov. Journ. Nucl. Phys.39, 998 (1984)

    work page 1984

  10. [10]

    M. H. Goroff and A. Sagnotti,The Ultraviolet Behavior of Einstein Gravity, Nucl. Phys. B266, 709–736 (1986)

    work page 1986

  11. [11]

    Faddeev-Popov ghosts in quantum gravity beyond perturbation theory

    A. Eichhorn,Faddeev-Popov ghosts in quantum gravity beyond perturbation theory, Phys. Rev. D87, 124016 (2013) [arXiv:1301.0632 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  12. [12]

    P. M. Lavrov and I. L. Shapiro,Gauge invariant renormalizability of quantum gravity, Phys. Rev. D100, 026018 (2019) [arXiv:1902.04687 [hep-th]]

    work page arXiv 2019

  13. [13]

    Mukhanov and S

    V. Mukhanov and S. Winitzki,Introduction to quantum effects in gravity, Cambridge University Press (UK), 2007

    work page 2007

  14. [14]

    R. P. Woodard,Perturbative Quantum Gravity Comes of Age, Int. J. Mod. Phys. D23, 1430020 (2014) [arXiv:1407.4748 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  15. [15]

    Kiefer,Quantum gravity, Oxford University Press (UK), 2017

    C. Kiefer,Quantum gravity, Oxford University Press (UK), 2017

    work page 2017

  16. [16]

    I. L. Buchbinder and I. L. Shapiro,Introduction to Quantum Field Theory with Applications to Quantum Gravity, Oxford University Press (UK), 2021

    work page 2021

  17. [17]

    Pagels,Energy-Momentum Structure Form Factors of Particles, Phys

    H. Pagels,Energy-Momentum Structure Form Factors of Particles, Phys. Rev.144, 1250–1260 (1966). 20

    work page 1966

  18. [18]

    Iwasaki,Quantum theory of gravitation vs

    Y. Iwasaki,Quantum theory of gravitation vs. classical theory. - fourth-order potential, Prog. Theor. Phys.46, 1587–1609 (1971)

    work page 1971

  19. [19]

    Hiida and H

    K. Hiida and H. Okamura,Gauge transformation and gravitational potentials, Prog. Theor. Phys47, 1743–1757 (1972)

    work page 1972

  20. [20]

    B. M. Barker and R. F. O’Connell,Gravitational Two-Body Problem with Arbitrary Masses, Spins, and Quadrupole Moments, Phys. Rev. D12, 329–335 (1975)

    work page 1975

  21. [21]

    J. F. Donoghue,Leading quantum correction to the newtonian potential, Phys. Rev. Lett.72, 2996 (1994) [arxiv:9310024 [gr-qc]]

    work page 1994

  22. [22]

    J. F. Donoghue,General relativity as an effective field theory: The leading quantum corrections, Phys. Rev. D50, 3874–3888 (1994) [arxiv:9405057 [gr-qc]]

    work page 1994

  23. [23]

    I. J. Muzinich and S. Vokos,Long range forces in quantum gravity, Phys. Rev. D52, 3472–3483 (1995) [arXiv:9501083 [hep-th]]

    work page 1995

  24. [24]

    H. W. Hamber and S. Liu,On the quantum corrections to the Newtonian potential, Phys. Lett. B357, 51–56 (1995) [arxiv:9505182 [hep-th]]

    work page 1995

  25. [25]

    N. E. J. Bjerrum-Bohr, J. F. Donoghue, and B. R. Holstein,Quantum corrections to the Schwarzschild and Kerr metrics, Phys. Rev. D,68084005 (2003) [arxiv:0211071 [hep-th]]

    work page 2003

  26. [26]

    N. E. J. Bjerrum Bohr, J. F. Donoghue, and B. R. Holstein,Quantum Gravitational Corrections to the Non- relativistic Scattering Potential of Two Masses, Phys. Rev. D67, 084033 (2003) [arxiv:0211072 [hep-th]]

    work page 2003

  27. [27]

    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. D102, no.8, 084015 (2020) [arXiv:2006.12824 [hep-th]]

    work page arXiv 2020

  28. [28]

    A. S. Majumder and S. Bhattacharya,Can aξRϕ 2 non-minimal coupling contribute to the non-relativistic gravitational potential?, [arXiv:2508.09523 [hep-th]]

    work page arXiv

  29. [29]

    J. F. Donoghue,Introduction to the effective field theory description of gravity(1995) [arXiv:9512024 [gr-qc]]

    work page 1995

  30. [30]

    A. A. Akhundov, S. Bellucci and A. Shiekh,Gravitational interaction to one loop in effective quantum gravity, Phys. Lett. B395, 16–23 (1997) [arxiv:9611018 [gr-qc]]

    work page 1997

  31. [31]

    N. E. J. Bjerrum-Bohr,Leading quantum gravitational corrections to scalar QED, Phys. Rev. D66, 084023 (2002) [arxiv:0206236 [hep-th]]

    work page 2002

  32. [32]

    J. F. Donoghue and T. Torma,Power counting of loop diagrams in general relativity, Physical Review D54, 4963-4972 (1996) [arXiv:9602121[hep-th]]

    work page 1996

  33. [33]

    Modanese,Potential energy in quantum gravity, Nucl

    G. Modanese,Potential energy in quantum gravity, Nucl. Phys. B434, 697–708 (1995) [arXiv:9408103 [hep-th]]

    work page 1995

  34. [34]

    Bern,Perturbative quantum gravity and its relation to gauge theory, Living Rev

    Z. Bern,Perturbative quantum gravity and its relation to gauge theory, Living Rev. Relt.5(2002) [arXiv:0206071 [gr-qc]]

    work page 2002

  35. [35]

    C. P. Burgess,Quantum gravity in everyday life: General relativity as an effective field theory, Living Rev. Rel.7, 5–56 (2004) [arXiv:0311082 [gr-qc]]

    work page 2004

  36. [36]

    W. D. Goldberger and I. Z. Rothstein,An Effective field theory of gravity for extended objects, Phys. Rev. D73, 104029 (2006) [arXiv:0409156 [hep-th]]. 21

    work page 2006

  37. [37]

    Akhundov and A

    A. Akhundov and A. Shiekh,A Review of Leading Quantum Gravitational Corrections to Newtonian Gravity, Electron. J. Theor. Phys.5, 1–16 (2008) [arXiv:0611091 [gr-qc]]

    work page 2008

  38. [38]

    B. R. Holstein and A. Ross,Spin Effects in Long Range Gravitational Scattering(2008) [arXiv:0802.0716 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  39. [39]

    N. E. J. Bjerrum-Bohr, J. F. Donoghue, B. R. Holstein, L. Plant´ e and P. Vanhove,Bending of Light in Quantum Gravity, Phys. Rev. Lett.114, 061301 (2015) [arXiv:1410.7590 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  40. [40]

    N. E. J. Bjerrum-Bohr, J. F. Donoghue, B. K. El-Menoufi, B. R. Holstein, L. Plant´ e and P. Vanhove,The Equivalence Principle in a Quantum World, Int. J. Mod. Phys. D24, 1544013 (2015) [arXiv:1505.04974 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  41. [41]

    Dong Bai and Yue Huang,More on the Bending of Light in Quantum Gravity, Phys. Rev. D95, 064045 (2017) [arXiv:1612.07629 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  42. [42]

    D. J. Toms,Quantum gravitational contributions to quantum electrodynamics, Nature468, 56–59 (2010) [arXiv:1010.0793 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  43. [43]

    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) [arXiv:1510.03291 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  44. [44]

    M. B. Fr¨ ob,Quantum gravitational corrections for spinning particles, JHEP10, 051 (2016) [arXiv:1607.03129 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  45. [45]

    Effective field theory approach to the gravitational two-body dynamics, at fourth post-Newtonian order and quintic in the Newton constant

    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. D95, 104009 (2017) [arXiv:1612.00482 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  46. [46]

    J. F. Donoghue, M. M. Ivanov and A. Shkerin,EPFL Lectures on General Relativity as a Quantum Field Theory, 2017 [arXiv:1702.00319 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  47. [47]

    S. C. Ulhoa, A. F. Santos, and F. C. Khanna,Scattering of fermions by gravitons, Gen. Relt. Gravit.49(2017) [arXiv:1608.00559 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  48. [48]

    Levi,Effective Field Theories of Post-Newtonian Gravity: A comprehensive review, Rept

    M. Levi,Effective Field Theories of Post-Newtonian Gravity: A comprehensive review, Rept. Prog. Phys.83, 075901 (2020) [arXiv:1807.01699 [hep-th]]

    work page arXiv 2020

  49. [49]

    Weak gravitational interaction of fermions: quantum viewpoint

    T. Olyaei, and A. Aziziy,Weak gravitational interaction of fermions: quantum viewpoint, Mod. Phys. Lett. A33, 1850218 (2018) [arXiv:1804.06939 [physics.gen-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  50. [50]

    M. B. Fr¨ ob,Graviton corrections to the Newtonian potential using invariant observables, JHEP01, 180 (2022) [arXiv:2109.09753 [hep-th]]

    work page arXiv 2022

  51. [51]

    Fierz and W

    M. Fierz and W. Pauli,On relativistic wave equations for particles of arbitrary spin in an electromagnetic field, Proc. R. Soc. Lond. A173, 211–232 (1939)

    work page 1939

  52. [52]

    Dvali,Black holes with quantum massive spin-2 hair, Phys

    G. Dvali,Black holes with quantum massive spin-2 hair, Phys. Rev. D74, 044013 (2006) [arXiv:0605295 [hep- th]]

    work page 2006

  53. [53]

    Claudia de Rham and Gregory Gabadadze,Selftuned Massive Spin-2, Phys. Lett. B693, 334–338 (2010) [arXiv:1006.4367 [hep-th]]. 22

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  54. [54]

    Generalization of the Fierz-Pauli Action

    C. de Rham and G. Gabadadze,Generalization of the Fierz-Pauli Action, Phys. Rev. D82, 044020, (2010) [arXiv:1007.0443 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  55. [55]

    Massive spin-2 theories

    S. Folkerts, C. Germani, and N. Wintergerst,Massive spin-2 theories, Cosmology and Particle Physics beyond Standard Models : Ten Years of the SEENET-MTP Network, 87–97 (2014) [arXiv:1310.0453 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  56. [56]

    Classical and quantum theory of the massive spin-two field

    A. Koenigstein, F. Giacosa, D. H. Rischke,Classical and quantum theory of the massive spin-two field, Ann. Phys.368, 16-55 (2016) [arXiv:1508.00110 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  57. [57]

    Jalali and A

    R. Jalali and A. Shirzad,Hamiltonian structure of Fierz-Pauli gravitons, partially massless fields and gauge symmetry, Nucl. Phys. B976, 115711 (2022) [arXiv:2010.04503 [hep-th]]

    work page arXiv 2022

  58. [58]

    J. A. Gill, D. Sengupta and A. G. Williams,Graviton-photon production with a massive spin-2 particle, Phys. Rev. D108, L051702 (2023) [arXiv:2303.04329 [hep-ph]]

    work page arXiv 2023

  59. [59]

    Farolfi and F

    L. Farolfi and F. Fecit,The Fierz-Pauli theory on curved spacetime at one-loop and its counterterms, Eur. Phys. J. C85, 356 (2025) [arXiv:2503.11304 [hep-th]]

    work page arXiv 2025

  60. [60]

    van Dam and M

    H. van Dam and M. J. G. Veltman,Massive and massless Yang-Mills and gravitational fields, Nucl. Phys. B22, 397 (1970)

    work page 1970

  61. [61]

    V. I. Zakharov,Linearized gravitation theory and the graviton mass, JETP Lett.12, 312 (1970)

    work page 1970

  62. [62]

    Ghosts in Massive Gravity

    P. Creminelli, A. Nicolis, M. Papucci, and E. Trincherini,Ghosts in massive gravity, JHEP0509, 003 (2005) [arXiv:hep-th/0505147]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  63. [63]

    Massive Gravity

    Claudia de Rham,Massive Gravity, Living Rev. Rel.17, 7 (2014) [arXiv:1401.4173 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  64. [64]

    Gambuti and N

    G. Gambuti and N. Maggiore,Fierz–Pauli theory reloaded: from a theory of a symmetric tensor field to linearized massive gravity, Eur. Phys. J. C81, 171 (2021) [arXiv:2102.10813 [gr-qc]]

    work page arXiv 2021

  65. [65]

    Tachinami and Y

    T. Tachinami and Y. Sendouda,Nonrelativistic stellar structure in the Fierz-Pauli theory and generic linear massive gravity, Phys. Rev. D110, 064013 (2024) [arXiv:2403.16363 [gr-qc]]. 23

    work page arXiv 2024