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WorksPhysics and Astronomy

Far-Zone Gravitational Interactions at the Sixth Post-Newtonian Order

Giacomo Brunello

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Abstract

We compute the nonlinear far-zone dynamics of compact binaries at the sixth post-Newtonian (6PN) order in the effective field theory of general relativity. We derive the complete O(G2)\mathcal{O}(G^2) far-zone action of a generic multipolar source in both the causal and the conservative prescriptions. Besides the known tails and failed tails, it contains five new sectors generated by the nonlinear memory of the emitted waves. On scattering orbits, they contain a boundary term, nonlocal in time, fixed by the balance of angular momentum. The same construction yields the emitted waveforms, whose energy and angular-momentum fluxes balance the losses of the action. For the binary, the 6PN dynamics also requires PN corrections to the lower-order sectors, the iterated radiation reaction, and a coupling quadratic in the curvature of the composite source, whose coefficient we fix by matching the gravitational Compton amplitude to that of the constituents. We compute the two-radiation contributions to the total scattering angle at fourth, fifth and sixth post-Minkowskian (PM) order and to the conservative angle at 4PM and 5PM, retaining the integer-PN terms through 6PN. For the conservative angle, we give both the local and γγ-3 prescriptions. Combined with the tail and the known potential contributions, our results reproduce the conservative and total 4PM angles and the second-self-force part of the conservative 5PM angle. Including the dissipative tail, we recover the total 5PM angle through 5PN and determine its new 6PN coefficient. At 6PM, the two-radiation contribution to the total angle extends the known 5PN result to 6PN.

Introduction

Gravitational waves from coalescing compact objects [1, 2, 3] carry a direct imprint of the relativistic two-body dynamics. The sensitivity of the next generation of detectors [4, 5, 6] increases the need for accurate predictions of the dynamics and radiation of compact binaries [7, 8, 9]. During the inspiral phase, the orbital velocity vv is small compared to the speed of light cc, and the dynamics admits the post-Newtonian (PN) expansion [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67]. An nnPN correction is suppressed by (v/c)2n(v/c)^{2n} relative to the Newtonian result. This phase is characterised by a separation of scales between the sizes of the compact objects, the orbital separation, and the wavelength of the emitted radiation. This separation of scales is organised by a tower of effective field theories (EFTs) [12, 14, 24, 68, 69, 70]: potential graviton modes mediate the near-zone orbital interaction, while radiation graviton modes propagate to infinity. In the far zone, the binary is described as a composite source with a mass, angular momentum and multipole moments [59, 60].

Multi-loop methods from quantum field theory have carried these EFTs to high PN orders [28, 31]. The complete binary dynamics has been determined at 4PN [16, 17, 20, 71, 72], also within the EFT approach [30, 31, 32, 33]. The 5PN impulse and scattering angle of the binary have recently been computed [73], combining the near-zone potential [37, 38, 40, 41] with the far-zone in-in action [42]. This also fixes the two coefficients left undetermined in the 5PN Hamiltonian of Ref. [74]. At 6PN, only partial results are known: the conservative potential up to O(G4)\mathcal{O}(G^4) [34, 43], the local- and nonlocal-in-time dynamics accessible through the self-force expansion [75, 76], and the static O(G7)\mathcal{O}(G^7) contribution [77], extended to O(G8)\mathcal{O}(G^8) at 7PN [78].

Complementary information comes from scattering orbits, where the dynamics can be expanded in GM/bG M/b, with MM the total mass and bb the impact parameter, retaining the full velocity dependence: this is the post-Minkowskian (PM) expansion [79, 80, 81, 82]. This expansion has been pushed to high orders with scattering-amplitude [83, 84, 85] and worldline [86, 87, 88, 89] methods. The conservative scattering angle is known at 3PM [90, 91, 92, 93, 94] and 4PM [95, 96, 97, 98, 99, 100]. Radiation-reaction effects on the scattering angle are known at 3PM [88, 101, 102, 103, 104, 105] and 4PM [106, 107, 108]. The radiated momentum is known at 3PM [88, 103, 109] and 4PM [106], and the loss of angular momentum through 3PM [101, 105, 110, 111] and at 4PM [112, 113, 114]. At 5PM, the conservative angle is known at first [115, 116, 117, 118] and second [119] self-force order, and in the potential region [120]. Its nonlocal-in-time tail contribution is known through tenth self-force order [121]. The radiation-reacted angle, the radiated energy and the recoil are known at 5PM at first self-force order [117]. In the PN expansion, radiation-reaction effects start at 2.5PN [122, 123, 124]. They were derived in the EFT framework in Ref. [125] and extended to 3.5PN [126, 127] and 4.5PN [128]. At higher orders, radiation interacts with the gravitational field of the source and with itself. These hereditary effects depend on the entire past history of the source. Scattering off the mass of the source produces the tail terms, which first enter the dynamics at 4PN [129, 130], and their iterations, the tails of tails, at 5.5PN [131]. Scattering off the angular momentum produces the failed tail terms [42, 132, 133, 134]. Interactions among the emitted waves generate gravitational-wave memory terms [130, 135, 136, 137, 138, 139]. In the multipolar post-Minkowskian (MPM) formalism, these effects are known in the flux and phasing of quasi-circular binaries through 4.5PN order and in the waveform modes through 4PN order [11, 21, 22]. In the EFT, causal forces and radiative observables are studied with the closed-time-path (in-in) formalism [125, 140, 141, 142, 143, 144]. The 4PN tail was derived in the EFT in Refs. [36, 145], and the 5PN hereditary terms were studied in Refs. [41, 132, 133, 146, 147, 148, 149]. The complete 5PN in-in effective action was obtained in Ref. [42], including the nonlocal boundary terms of the memory contribution. In the far zone at 6PN only the tails and failed tails are known for general multipoles [134, 150].

In this work, we compute the nonlinear far-zone contribution to the dynamics of two non-spinning point masses at 6PN. We first derive, from a Berends–Giele recursion for the in-in one-point function, the complete O(G2)O(G^2) far-zone action of a generic source, which contains eleven sectors: two tails, four failed tails and five memories. We reproduce the known tail and failed-tail results [134, 150] and compute the five memory actions in both the causal and the conservative prescriptions. On scattering orbits, they contain in addition a boundary term, nonlocal in time, fixed by the balance of angular momentum, in which the nonlinear memory of the emitted waves [130, 135, 136, 137] acts on the change of the mass multipoles. From the same generating functional we compute the nonlinear waveforms, whose stress tensor is conserved up to the recoil of the source. We verify that the radiated energy and angular momentum balance the losses predicted by the action at generic frequencies, while the zero-frequency part of the angular-momentum flux reproduces the coefficients of the boundary term.

The far-zone action of a generic source is not yet the far-zone contribution to the binary dynamics. The multipoles are matched to the near-zone dynamics [59, 60], the lower-order far-zone sectors are evaluated with PN-corrected multipoles and motion [66, 76, 146, 151, 152, 153], and the radiation-reaction forces are iterated in the equations of motion, which generates conservative-like contributions at second order in the leading force [42]. A coupling quadratic in the electric curvature of the composite source also enters, which the covariant multipolar couplings do not generate. We fix its coefficient by matching the gravitational Compton amplitude of the multipoles to that of the constituents.

We compute the two-radiation contributions to the total scattering angle at 4PM, 5PM and 6PM and to the conservative angle at 4PM and 5PM, retaining the integer-PN terms through 6PN. The conservative results are given in both the local and γ\gamma-3 prescriptions [73, 119].

At 4PM, combining the two-radiation contributions with the tail evaluated in dimensional regularisation and the known potential [33, 43], we reproduce both the conservative and the total angle [96, 99, 106] through 6PN.

At 5PM, the conservative angle depends on the prescription used to separate it, while the total angle does not. After adding the potential region [120] and the conservative tail, the local result agrees with Ref. [73] at 5PN and extends it to 6PN. The γ\gamma-3 result agrees with Ref. [119] at second self-force order through 6PN. In both prescriptions, the cubic terms in the symmetric mass ratio ν\nu cancel. Combining the retarded two-radiation contribution with the potential and tail contributions, including the dissipative tail, gives the total 5PM angle at orders ν2\nu^2 and ν3\nu^3 through 6PN. The 5PN result agrees with Ref. [73], while the 6PN ν2\nu^2 coefficient is new and the ν3\nu^3 coefficient agrees with Ref. [117].

At 6PM, we give the two-radiation contribution to the total angle. It reproduces the 5PN result of Ref. [73] and extends it to 6PN.

Conventions. We work in dimensional regularisation with D=d+1D=d+1 space-time dimensions, d=3−2εd=3-2\varepsilon, in the mostly-minus signature ημν=diag⁡(+,…,−)\eta_{\mu\nu}=\operatorname{diag}(+,\ldots,-), and we set c=1c=1. The gravitational coupling is denoted by κ=32πG\kappa=\sqrt{32\pi G}, with GG Newton’s constant. We use shorthand notation for integrals

∫k≡μ2ε∫dDk(2π)D,∫p≡μ2ε∫ddp(2π)d,(1)\int_k \equiv\mu^{2\varepsilon}\int\frac{\mathrm{d}^D k}{(2\pi)^D}, \qquad\int_p \equiv\mu^{2\varepsilon}\int\frac{\mathrm{d}^d p}{(2\pi)^d}, \tag*{(1)}

where bold notation denotes spatial loop momenta, and μ\mu is the scale introduced by dimensional regularisation. We define the scale μˉ\bar{\mu} as

μˉ2=μ2πe−γE.(2)\bar{\mu}^2=\mu^2\pi e^{-\gamma_E}. \tag*{(2)}

Fourier transforms are defined as

hμν(x)=∫ke−ik⋅xhμν(k),IL(t)=∫dω2πe−iωtIL(ω).(3)h_{\mu\nu}(x)=\int_k e^{-ik\cdot x}h_{\mu\nu}(k), \qquad\mathcal{I}^{L}(t)=\int\frac{\mathrm{d}\omega}{2\pi}e^{-i\omega t}\mathcal{I}^{L}(\omega). \tag*{(3)}

Multi-index notation L=i1⋯iℓL=i_1\cdots i_\ell is used for the symmetric trace-free (STF) multipoles, and spatial indices are contracted with the Euclidean metric. Angular brackets on indices, as in X⟨L⟩X^{\langle L\rangle}, or a superscript STF on a product denote the STF projection. Time derivatives are denoted by dots or by superscripts, X(n)=dnX/dtnX^{(n)}=\mathrm{d}^nX/\mathrm{d}t^n, and ΔX≡X(+∞)−X(−∞)\Delta X\equiv X(+\infty)-X(-\infty) is the change of a quantity over a scattering history.

Far-zone effective theory and in-in observables

The far-zone theory describes the binary as a multipolar source coupled to the radiation field. We specify its action and Feynman rules, then construct the causal in-in effective action and the one-point function using recursion relations. We introduce power counting relations to identify the diagrams relevant at 6PN order.

From the binary to the far-zone source

The inspiral phase of a compact binary coalescence is characterised by a hierarchy of length scales,

R≪r≪λGW,(4)R \ll r \ll\lambda_{\mathrm{GW}}, \tag*{(4)}

where RR is the characteristic size of each compact object, rr the orbital separation and λGW\lambda_{\mathrm{GW}} the radiation wavelength. For compact objects, R∼GM/c2R \sim GM/c^2, and the virial theorem for bound orbits gives v2/c2∼GM/(rc2)v^2/c^2 \sim GM/(rc^2), so that R/r∼v2/c2R/r \sim v^2/c^2. Since the radiation is emitted at the orbital frequency, r/λGW∼v/cr/\lambda_{\mathrm{GW}} \sim v/c. Hence, all scales are related by the small parameter v/c≪1v/c \ll1, which defines the post-Newtonian (PN) expansion. This hierarchy allows the problem to be studied with a tower of EFTs, one for each scale [24, 59] (see Refs. [12, 14, 70] for comprehensive reviews). At distances larger than RR, each body is represented by a worldline, with its internal structure encoded in finite-size operators. At the orbital scale rr, the two worldlines interact through the exchange of gravitational fields, described by the action

Snear=Sbulk+∑a=12Spp(a).(5)S_{\mathrm{near}} = S_{\mathrm{bulk}} + \sum_{a=1}^{2} S_{\mathrm{pp}}^{(a)} . \tag*{(5)}

where Spp(a)S_{\mathrm{pp}}^{(a)} is the point-particle action

Spp(a)=−ma∫dλgμνuaμuaν,(6)S_{\mathrm{pp}}^{(a)} = -m_a \int\mathrm{d}\lambda\sqrt{g_{\mu\nu}u_a^\mu u_a^\nu}, \tag*{(6)}

with mam_a the mass of body aa, λ\lambda a worldline parameter and uaμ=dxaμ/dλu_a^\mu= \mathrm{d}x_a^\mu/\mathrm{d}\lambda its four-velocity. The bulk gravitational dynamics is governed by the Einstein–Hilbert action with the harmonic gauge-fixing term,

Sbulk=SEH+Sgf=∫dDx[−2κ2−g R+1κ2GμGμ],(7)S_{\mathrm{bulk}} = S_{\mathrm{EH}} + S_{\mathrm{gf}} = \int\mathrm{d}^{D}x \left[-\frac{2}{\kappa^2}\sqrt{-g}\,R + \frac{1}{\kappa^2}G_\mu G^\mu\right], \tag*{(7)}

where Gμ=κ(∂αhαμ−12∂μh)G_\mu= \kappa(\partial^\alpha h_{\alpha\mu} - \frac{1}{2}\partial_\mu h) is written in terms of the graviton field, defined by expanding the metric around flat space-time,

gμν=ημν+κhμν.(8)g_{\mu\nu} = \eta_{\mu\nu} + \kappa h_{\mu\nu}. \tag*{(8)}

The graviton modes do not scale homogeneously with the velocity vv. Potential modes HμνH_{\mu\nu}, with momenta kμ∼(v/r,1/r)k^\mu\sim(v/r,1/r), are off shell and mediate the orbital interaction. Radiation modes hˉμν\bar{h}_{\mu\nu}, with momenta kμ∼(v/r,v/r)k^\mu\sim(v/r,v/r), propagate on shell to infinity,

hμν=Hμν+hˉμν.(9)h_{\mu\nu} = H_{\mu\nu} + \bar{h}_{\mu\nu}. \tag*{(9)}

This separation implements the method of regions [154] in the two-body problem [24], and is well defined in dimensional regularisation, where the scaleless integrals that would mix the regions vanish. The two effective theories that result are tied to each other by the cancellation of the ultraviolet poles of the far zone against the infrared poles of the near zone [27, 32, 33, 71].

Far-zone source action

We integrate out the potential modes while retaining the orbital variables and treating the radiation field as a background. Since kradr∼v≪1k_{\mathrm{rad}}r \sim v \ll1, the radiation field does not resolve the orbital scale, and its coupling is organised as a multipole expansion about the centre of mass,

eiSeff[xa,hˉ]=∫DH eiSnear[xa,H+hˉ],e^{iS_{\mathrm{eff}}[x_a,\bar{h}]} = \int\mathcal{D}H\,e^{iS_{\mathrm{near}}[x_a,H+\bar{h}]},

References

  1. [1]B. P. Abbott et al. (LIGO Scientific, Virgo), Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett. 116, 061102 (2016), arXiv:1602.03837 [gr-qc].
  2. [2]R. Abbott et al. (KAGRA, VIRGO, LIGO Scientific), GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo during the Second Part of the Third Observing Run, Phys. Rev. X 13, 041039 (2023), arXiv:2111.03606 [gr-qc].DOI
  3. [3]A. G. Abac et al. (LIGO Scientific, VIRGO, KAGRA), GWTC-4.0: Updating the Gravitational-Wave Transient Catalog with Observations from the First Part of the Fourth LIGO-Virgo-KAGRA Observing Run, Astrophys. J. Lett. 104, L22 (2026), arXiv:2508.18082 [gr-qc].DOI
  4. [4]M. Punturo et al., The Einstein Telescope: A third-generation gravitational wave observatory, Class. Quant. Grav. 27, 194002 (2010).
  5. [5]D. Reitze et al., Cosmic Explorer: The U.S. Contribution to Gravitational-Wave Astronomy beyond LIGO, Bull. Am. Astron. Soc. 51, 035 (2019), arXiv:1907.04833 [astro-ph.IM].arxiv.org/abs/1907.04833
  6. [6]P. Amaro-Seoane et al. (LISA), Laser Interferometer Space Antenna (2017), arXiv:1702.00786 [astro-ph.IM].
  7. [7]M. Pürrer and C.-J. Haster, Gravitational waveform accuracy requirements for future ground-based detectors, Phys. Rev. Res. 2, 023151 (2020), arXiv:1912.10055 [gr-qc].arxiv.org/abs/1912.10055
  8. [8]C. B. Owen, C.-J. Haster, S. Perkins, N. J. Cornish, and N. Yunes, Waveform accuracy and systematic uncertainties in current gravitational wave observations, Phys. Rev. D 108, 044018 (2023), arXiv:2301.11941 [gr-qc].DOI
  9. [9]A. Dhani, S. H. Völkel, A. Buonanno, H. Estelles, J. Gair, H. P. Pfeiffer, L. Pompili, and A. Toubaiana, Systematic Biases in Estimating the Properties of Black Holes Due to Inaccurate Gravitational-Wave Models, Phys. Rev. X 15, 031036 (2025), arXiv:2404.05811 [gr-qc].DOI
  10. [10]A. Einstein, L. Infeld, and B. Hoffmann, The Gravitational equations and the problem of motion, Annals Math. 39, 65 (1938).DOI
  11. [11]L. Blanchet, Post-Newtonian Theory for Gravitational Waves, Living Rev. Rel. 17, 2 (2014), arXiv:1310.1528 [gr-qc].arxiv.org/abs/1310.1528
  12. [12]R. A. Porto, The effective field theorist’s approach to gravitational dynamics, Phys. Rept. 633, 1 (2016), arXiv:1601.04914 [hep-th].DOI
  13. [13]G. Schäfer and P. Jaranowski, Hamiltonian formulation of general relativity and post-Newtonian dynamics of compact binaries, Living Rev. Rel. 27, 2 (2024), arXiv:1805.07240 [gr-qc].
  14. [14]M. Levi, Effective Field Theories of Post-Newtonian Gravity: A comprehensive review, Rept. Prog. Phys. 83, 075901 (2020), arXiv:1807.01699 [hep-th].DOI
  15. [15]P. Jaranowski and G. Schäfer, Third post-Newtonian higher order ADM Hamilton dynamics for two-body point mass systems, Phys. Rev. D 57, 7274 (1998), [Erratum: Phys.Rev.D 63, 029902 (2001)], arXiv:gr-qc/9712075.
  16. [16]T. Damour, P. Jaranowski, and G. Schäfer, Nonlocal-in-time action for the fourth post-Newtonian conservative dynamics of two-body systems, Phys. Rev. D 89, 064058 (2014), arXiv:1401.4548 [gr-qc].DOI
  17. [17]P. Jaranowski and G. Schäfer, Derivation of local-in-time fourth post-Newtonian ADM Hamiltonian for spinless compact binaries, Phys. Rev. D 92, 124043 (2015), arXiv:1508.01016 [gr-qc].DOI
  18. [18]L. Bernard, L. Blanchet, A. Bohé, G. Faye, and S. Marsat, Fokker action of nonspinning compact binaries at the fourth post-Newtonian approximation, Phys. Rev. D 93, 084037 (2016), arXiv:1512.02876 [gr-qc].DOI
  19. [19]L. Bernard, L. Blanchet, A. Bohé, G. Faye, and S. Marsat, Energy and periastron advance of compact binaries on circular orbits at the fourth post-Newtonian order, Phys. Rev. D 95, 044026 (2017), arXiv:1610.07934 [gr-qc].DOI
  20. [20]T. Damour, P. Jaranowski, and G. Schäfer, Conservative dynamics of two-body systems at the fourth post-Newtonian approximation of general relativity, Phys. Rev. D 93, 084014 (2016), arXiv:1601.01283 [gr-qc].DOI
  21. [21]L. Blanchet, G. Faye, Q. Henry, F. Larrouturou, and D. Trestini, Gravitational-wave flux and quadrupole modes from quasicircular nonspinning compact binaries to the fourth post-Newtonian order, Phys. Rev. D 108, 064041 (2023), arXiv:2304.11186 [gr-qc].DOI
  22. [22]L. Blanchet, G. Faye, Q. Henry, F. Larrouturou, and D. Trestini, Gravitational-Wave Phasing of Quasicircular Compact Binary Systems to the Fourth-and-a-Half Post-Newtonian Order, Phys. Rev. Lett. 131, 121402 (2023), arXiv:2304.11185 [gr-qc].DOI
  23. [23]T. Damour and P. Jaranowski, Four-loop static contribution to the gravitational interaction potential of two point masses, Phys. Rev. D 95, 084005 (2017), arXiv:1701.02645 [gr-qc].DOI
  24. [24]W. D. Goldberger and I. Z. Rothstein, An Effective field theory of gravity for extended objects, Phys. Rev. D 73, 104029 (2006), arXiv:hep-th/0409156.DOI
  25. [25]J. B. Gilmore and A. Ross, Effective field theory calculation of second post-Newtonian binary dynamics, Phys. Rev. D 78, 124021 (2008), arXiv:0810.1328 [gr-qc].DOI
  26. [26]S. Foffa and R. Sturani, Effective field theory calculation of conservative binary dynamics at third post-Newtonian order, Phys. Rev. D 84, 044031 (2011), arXiv:1104.1122 [gr-qc].DOI
  27. [27]R. A. Porto and I. Z. Rothstein, Apparent ambiguities in the post-Newtonian expansion for binary systems, Phys. Rev. D 96, 024062 (2017), arXiv:1703.06433 [gr-qc].DOI
  28. [28]B. Kol and R. Shir, Classical 3-loop 2-body diagrams, JHEP 09, 069, arXiv:1306.3220 [hep-th].
  29. [29]S. Foffa and R. Sturani, Dynamics of the gravitational two-body problem at fourth post-Newtonian order and at quadratic order in the Newton constant, Phys. Rev. D 87, 064011 (2013), arXiv:1206.7087 [gr-qc].DOI
  30. [30]S. Foffa and R. Sturani, Conservative dynamics of binary systems to fourth Post-Newtonian order in the EFT approach I: Regularized Lagrangian, Phys. Rev. D 100, 024047 (2019), arXiv:1903.05113 [gr-qc].DOI
  31. [31]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), arXiv:1612.00482 [gr-qc].DOI
  32. [32]S. Foffa, R. A. Porto, I. Rothstein, and R. Sturani, Conservative dynamics of binary systems to fourth Post-Newtonian order in the EFT approach II: Renormal- ized Lagrangian Phys. Rev. D 100, 024048 (2019), arXiv:1903.05118 [gr-qc].
  33. [33]J. Blümlein, A. Maier, P. Marquard, and G. Schäfer, Fourth post-Newtonian Hamiltonian dynamics of two-body systems from an effective field theory approach, Nucl. Phys. B 955, 115041 (2020), arXiv:2003.01692 [gr-qc].DOI
  34. [34]J. Blümlein, A. Maier, P. Marquard, and G. Schäfer, Testing binary dynamics in gravity at the sixth post-Newtonian level, Phys. Lett. B 807, 135496 (2020), arXiv:2003.07145 [gr-qc].DOI
  35. [35]W. D. Goldberger, A. Ross, and I. Z. Rothstein, Black hole mass dynamics and renormalization group evolution, Phys. Rev. D 89, 124033 (2014), arXiv:1211.6095 [hep-th].DOI
  36. [36]C. R. Galley, A. K. Leibovich, R. A. Porto, and A. Ross, Tail effect in gravitational radiation reaction: Time non-locality and renormalization group evolution, Phys. Rev. D 93, 124010 (2016), arXiv:1511.07379 [gr-qc].DOI
  37. [37]S. Foffa, P. Mastrolia, R. Sturani, C. Sturm, and W. J. Torres Bobadilla, Static two-body potential at fifth post-Newtonian order, Phys. Rev. Lett. 122, 241605 (2019), arXiv:1902.10571 [gr-qc].DOI
  38. [38]J. Blümlein, A. Maier, and P. Marquard, Five-Loop Static Contribution to the Gravitational Interaction Potential of Two Point Masses, Phys. Lett. B 800, 135100 (2020), arXiv:1902.11180 [gr-qc].DOI
  39. [39]S. Foffa, R. Sturani, and W. J. Torres Bobadilla, Efficient resummation of high post-Newtonian contributions to the binding energy, JHEP 02, 165, arXiv:2010.13730 [gr-qc].DOI
  40. [40]J. Blümlein, A. Maier, P. Marquard, and G. Schäfer, The fifth-order post-Newtonian Hamiltonian dynamics of two-body systems from an effective field theory approach: potential contributions, Nucl. Phys. B 965, 115352 (2021), arXiv:2010.13672 [gr-qc].DOI
  41. [41]J. Blümlein, A. Maier, P. Marquard, and G. Schäfer, The fifth-order post-Newtonian Hamiltonian dynamics of two-body systems from an effective field theory approach, Nucl. Phys. B 983, 115900 (2022), [Erratum: Nucl.Phys.B 985, 115991 (2022)], arXiv:2110.13822 [gr-qc].DOI
  42. [42]R. A. Porto, M. M. Riva, and Z. Yang, Nonlinear gravitational radiation reaction: failed tail, memories & squares, JHEP 04, 050, arXiv:2409.05860 [gr-qc].arxiv.org/abs/2409.05860
  43. [43]J. Blümlein, A. Maier, P. Marquard, and G. Schäfer, The 6th post-Newtonian potential terms at O(G⁴_N), Phys. Lett. B 816, 136260 (2021), arXiv:2101.08630 [gr-qc].DOI
  44. [44]R. A. Porto, Post-Newtonian corrections to the motion of spinning bodies in NRGR, Phys. Rev. D 73, 104031 (2006), arXiv:gr-qc/0511061.DOI
  45. [45]R. A. Porto and I. Z. Rothstein, The Hyperfine Einstein-Infeld-Hoffmann potential, Phys. Rev. Lett. 97, 021101 (2006), arXiv:gr-qc/0604099.DOI
  46. [46]R. A. Porto and I. Z. Rothstein, Spin(1)Spin(2) Effects in the Motion of Inspiralling Compact Binaries at Third Order in the Post-Newtonian Expansion, Phys. Rev. D 78, 044012 (2008), [Erratum: Phys.Rev.D 81, 029904 (2010)], arXiv:0802.0720 [gr-qc].arxiv.org/abs/0802.0720
  47. [47]M. Levi, Next to Leading Order gravitational Spin1-Spin2 coupling with Kaluza-Klein reduction, Phys. Rev. D 82, 064029 (2010), arXiv:0802.1508 [gr-qc].DOI
  48. [48]M. Levi and J. Steinhoff, Spinning gravitating objects in the effective field theory in the post-Newtonian scheme, JHEP 09, 219, arXiv:1501.04956 [gr-qc].
  49. [49]M. Levi and J. Steinhoff, Complete conservative dynamics for inspiralling compact binaries with spins at the fourth post-Newtonian order, JCAP 09, 029, arXiv:1607.04252 [gr-qc].DOI
  50. [50]J.-W. Kim, M. Levi, and Z. Yin, Quadratic-in-spin interactions at fifth post-Newtonian order: probe new physics, Phys. Lett. B 834, 137410 (2022), arXiv:2112.01509 [hep-th].DOI
  51. [51]M. Levi, A. J. McLeod, and M. Von Hippel, N³LO gravitational quadratic-in-spin interactions at G⁴, JHEP 07, 116, arXiv:2003.07890 [hep-th].
  52. [52]M. Levi, S. Mougiakakos, and M. Vieira, Gravitational cubic-in-spin interaction at the next-to-leading post-Newtonian order, JHEP 01, 036, arXiv:1912.06276 [hep-th].DOI
  53. [53]J.-W. Kim, M. Levi, and Z. Yin, N³LO spin-orbit interaction via the EFT of spinning gravitating objects, JHEP 05, 184, arXiv:2208.14949 [hep-th].DOI
  54. [54]J.-W. Kim, M. Levi, and Z. Yin, N³LO quadratic-in-spin interactions for generic compact binaries, JHEP 03, 098, arXiv:2209.09235 [hep-th].DOI
  55. [55]M. Levi, R. Morales, and Z. Yin, From the EFT of spinning gravitating objects to Poincaré and gauge invariance at the 4.5PN precision frontier, JHEP 09, 090, arXiv:2210.17538 [hep-th].DOI
  56. [56]M. Levi and Z. Yin, Completing the fifth PN precision frontier via the EFT of spinning gravitating objects, JHEP 04, 079, arXiv:2211.14018 [hep-th].DOI
  57. [57]M. K. Mandal, P. Mastrolia, R. Patil, and J. Steinhoff, Gravitational spin-orbit Hamiltonian at NNNLO in the post-Newtonian framework, JHEP 03, 130, arXiv:2209.00611 [hep-th].
  58. [58]M. K. Mandal, P. Mastrolia, R. Patil, and J. Steinhoff, Gravitational quadratic-in-spin Hamiltonian at NNNLO in the post-Newtonian framework, JHEP 07, 128, arXiv:2210.09176 [hep-th].
  59. [59]W. D. Goldberger and A. Ross, Gravitational radiative corrections from effective field theory, Phys. Rev. D 81, 124015 (2010), arXiv:0912.4254 [gr-qc].DOI
  60. [60]A. Ross, Multipole expansion at the level of the action, Phys. Rev. D 85, 125033 (2012), arXiv:1202.4750 [gr-qc].DOI
  61. [61]R. A. Porto, A. Ross, and I. Z. Rothstein, Spin induced multipole moments for the gravitational wave flux from binary inspirals to third Post-Newtonian order, JCAP 03, 009, arXiv:1007.1312 [gr-qc].DOI
  62. [62]N. T. Maia, C. R. Galley, A. K. Leibovich, and R. A. Porto, Radiation reaction for spinning bodies in effective field theory I: Spin-orbit effects, Phys. Rev. D 96, 084064 (2017), arXiv:1705.07934 [gr-qc].DOI
  63. [63]G. Cho, B. Pardo, and R. A. Porto, Gravitational radiation from inspiralling compact objects: Spin-spin effects completed at the next-to-leading post-Newtonian order, Phys. Rev. D 104, 024037 (2021), arXiv:2103.14612 [gr-qc].DOI
  64. [64]G. Cho, R. A. Porto, and Z. Yang, Gravitational radiation from inspiralling compact objects: Spin effects to the fourth post-Newtonian order, Phys. Rev. D 106, L101501 (2022), arXiv:2201.05138 [gr-qc].DOI
  65. [65]L. Amalberti, F. Larrouturou, and Z. Yang, Multipole expansion at the level of the action in d-dimensions, Phys. Rev. D 109, 104027 (2024), arXiv:2312.02868 [gr-qc].DOI
  66. [66]L. Amalberti, Z. Yang, and R. A. Porto, Gravitational radiation from inspiralling compact binaries to N3LO in the effective field theory approach, Phys. Rev. D 110, 044046 (2024), arXiv:2406.03457 [gr-qc].
  67. [67]M. K. Mandal, P. Mastrolia, R. Patil, and J. Steinhoff, Radiating Love: adiabatic tidal fluxes and modes up to next-to-next-to-leading post-Newtonian order, JHEP 05, 008, arXiv:2412.01706 [gr-qc].DOI
  68. [68]W. D. Goldberger, Les Houches lectures on effective field theories and gravitational radiation, in Les Houches Summer School - Session 86: Particle Physics and Cosmology: The Fabric of Spacetime (2007) arXiv:hep-ph/0701129.
  69. [69]S. Foffa and R. Sturani, Effective field theory methods to model compact binaries, Class. Quant. Grav. 31, 043001 (2014), arXiv:1309.3474 [gr-qc].DOI
  70. [70]W. D. Goldberger, Effective Field Theory for Compact Binary Dynamics (2022), arXiv:2212.06677 [hep-th].arxiv.org/abs/2212.06677
  71. [71]L. Bernard, L. Blanchet, A. Bohé, G. Faye, and S. Marsat, Dimensional regularization of the IR divergences in the Fokker action of point-particle binaries at the fourth post-Newtonian order, Phys. Rev. D 96, 104043 (2017), arXiv:1706.08480 [gr-qc].DOI
  72. [72]T. Marchand, L. Bernard, L. Blanchet, and G. Faye, Ambiguity-Free Completion of the Equations of Motion of Compact Binary Systems at the Fourth Post-Newtonian Order, Phys. Rev. D 97, 044023 (2018), arXiv:1707.09289 [gr-qc].DOI
  73. [73]R. A. Porto and M. M. Riva, Black hole dynamics at fifth post-Newtonian order, JHEP 09, 075, arXiv:2604.09545 [gr-qc].
  74. [74]D. Bini, T. Damour, and A. Geralico, Binary dynamics at the fifth and fifth-and-a-half post-Newtonian orders, Phys. Rev. D 102, 024062 (2020), arXiv:2003.11891 [gr-qc].DOI
  75. [75]D. Bini, T. Damour, and A. Geralico, Sixth post-Newtonian local-in-time dynamics of binary systems, Phys. Rev. D 102, 024061 (2020), arXiv:2004.05407 [gr-qc].DOI
  76. [76]D. Bini, T. Damour, and A. Geralico, Sixth post-Newtonian nonlocal-in-time dynamics of binary systems, Phys. Rev. D 102, 084047 (2020), arXiv:2007.11239 [gr-qc].DOI
  77. [77]G. Brunello, M. K. Mandal, P. Mastrolia, R. Patil, M. Pegorin, J. Ronca, S. Smith, J. Steinhoff, and W. J. Torres Bobadilla, Six-Loop Gravitational Interactions at the Sixth Post-Newtonian Order, Phys. Rev. Lett. 137, 111401 (2026), arXiv:2512.19498 [hep-th].DOI
  78. [78]G. Brunello, M. K. Mandal, P. Mastrolia, R. Patil, M. Pegorin, S. Smith, and J. Steinhoff, All-order structure of static gravitational interactions and the seventh post-Newtonian potential (2026), arXiv:2604.14134 [hep-th].arxiv.org/abs/2604.14134
  79. [79]K. Westpfahl and M. Goller, Gravitational scattering of two relativistic particles in postlinear approximation, Lett. Nuovo Cim. 26, 573 (1979).DOI
  80. [80]K. Westpfahl, High-Speed Scattering of Charged and Uncharged Particles in General Relativity, Fortsch. Phys. 33, 417 (1985).
  81. [81]T. Damour, Gravitational scattering, post-Minkowskian approximation and Effective One-Body theory, Phys. Rev. D 94, 104015 (2016), arXiv:1609.00354 [gr-qc].DOI
  82. [82]T. Damour, High-energy gravitational scattering and the general relativistic two-body problem, Phys. Rev. D 97, 044038 (2018), arXiv:1710.10599 [gr-qc].DOI
  83. [83]N. E. J. Bjerrum-Bohr, P. H. Damgaard, G. Festuccia, L. Planté, and P. Vanhove, General Relativity from Scattering Amplitudes, Phys. Rev. Lett. 121, 171601 (2018), arXiv:1806.04920 [hep-th].
  84. [84]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), arXiv:1808.02489 [hep-th].DOI
  85. [85]D. A. Kosower, B. Maybee, and D. O’Connell, Amplitudes, Observables, and Classical Scattering, JHEP 02, 137, arXiv:1811.10950 [hep-th].DOI
  86. [86]G. Kälin and R. A. Porto, Post-Minkowskian Effective Field Theory for Conservative Binary Dynamics, JHEP 11, 106, arXiv:2006.01184 [hep-th].DOI
  87. [87]G. Mogull, J. Plefka, and J. Steinhoff, Classical black hole scattering from a worldline quantum field theory, JHEP 02, 048, arXiv:2010.02865 [hep-th].
  88. [88]G. Kälin, J. Neef, and R. A. Porto, Radiation-reaction in the Effective Field Theory approach to Post-Minkowskian dynamics, JHEP 01, 140, arXiv:2207.00580 [hep-th].DOI
  89. [89]G. U. Jakobsen, G. Mogull, J. Plefka, and B. Sauer, All things related: radiation-reaction in worldline quantum field theory, JHEP 10, 128, arXiv:2207.00569 [hep-th].arxiv.org/abs/2207.00569
  90. [90]Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M. P. Solon, and M. Zeng, Scattering Amplitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order, Phys. Rev. Lett. 122, 201603 (2019), arXiv:1901.04424 [hep-th].DOI
  91. [91]Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M. P. Solon, and M. Zeng, Black Hole Binary Dynamics from the Double Copy and Effective Theory, JHEP 10, 206, arXiv:1908.01493 [hep-th].DOI
  92. [92]C. Cheung and M. P. Solon, Classical gravitational scattering at O(G^3) from Feynman diagrams, JHEP 06, 144, arXiv:2003.08351 [hep-th].DOI
  93. [93]J. Parra-Martinez, M. S. Ruf, and M. Zeng, Extremal black hole scattering at O(G^3): graviton dominance, eikonal exponentiation, and differential equations, JHEP 11, 023, arXiv:2005.04236 [hep-th].
  94. [94]G. Kälin, Z. Liu, and R. A. Porto, Conservative Dynamics of Binary Systems to Third Post-Minkowskian Order from the Effective Field Theory Approach, Phys. Rev. Lett. 125, 261103 (2020), arXiv:2007.04977 [hep-th].DOI
  95. [95]Z. Bern, J. Parra-Martinez, R. Roiban, M. S. Ruf, C.-H. Shen, M. P. Solon, and M. Zeng, Scattering Amplitudes and Conservative Binary Dynamics at O(G^4), Phys. Rev. Lett. 126, 171601 (2021), arXiv:2101.07254 [hep-th].DOI
  96. [96]Z. Bern, J. Parra-Martinez, R. Roiban, M. S. Ruf, C.-H. Shen, M. P. Solon, and M. Zeng, Scattering Amplitudes, the Tail Effect, and Conservative Binary Dynamics at O(G4), Phys. Rev. Lett. 128, 161103 (2022), arXiv:2112.10750 [hep-th].DOI
  97. [97]Z. Bern, J. Parra-Martinez, R. Roiban, M. S. Ruf, C.-H. Shen, M. P. Solon, and M. Zeng, Scattering amplitudes and conservative dynamics at the fourth post-Minkowskian order, PoS LL2022, 051 (2022).
  98. [98]C. Dlapa, G. Kälin, Z. Liu, and R. A. Porto, Dynamics of binary systems to fourth Post-Minkowskian order from the effective field theory approach, Phys. Lett. B 831, 137203 (2022), arXiv:2106.08276 [hep-th].DOI
  99. [99]C. Dlapa, G. Kälin, Z. Liu, and R. A. Porto, Conservative Dynamics of Binary Systems at Fourth Post-Minkowskian Order in the Large-Eccentricity Expansion, Phys. Rev. Lett. 128, 161104 (2022), arXiv:2112.11296 [hep-th].DOI
  100. [100]C. Dlapa, G. Kälin, Z. Liu, and R. A. Porto, Local in Time Conservative Binary Dynamics at Fourth Post-Minkowskian Order, Phys. Rev. Lett. 132, 221401 (2024), arXiv:2403.04853 [hep-th].arxiv.org/abs/2403.04853
  101. [101]T. Damour, Radiative contribution to classical gravitational scattering at the third order in G, Phys. Rev. D 102, 124008 (2020), arXiv:2010.01641 [gr-qc].DOI
  102. [102]P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, The eikonal approach to gravitational scattering and radiation at O(G^3), JHEP 07, 169, arXiv:2104.02856 [hep-th].
  103. [103]E. Herrmann, J. Parra-Martinez, M. S. Ruf, and M. Zeng, Radiative classical gravitational observables at O(G^3) from scattering amplitudes, JHEP 10, 148, arXiv:2104.03957 [hep-th].DOI
  104. [104]D. Bini, T. Damour, and A. Geralico, Radiative contributions to gravitational scattering, Phys. Rev. D 104, 084031 (2021), arXiv:2107.08896 [gr-qc].DOI
  105. [105]A. V. Manohar, A. K. Ridgway, and C.-H. Shen, Radiated Angular Momentum and Dissipative Effects in Classical Scattering, Phys. Rev. Lett. 129, 121601 (2022), arXiv:2203.04283 [hep-th].DOI
  106. [106]C. Dlapa, G. Kälin, Z. Liu, J. Neef, and R. A. Porto, Radiation Reaction and Gravitational Waves at Fourth Post-Minkowskian Order, Phys. Rev. Lett. 130, 101401 (2023), arXiv:2210.05541 [hep-th].DOI
  107. [107]G. U. Jakobsen, G. Mogull, J. Plefka, and B. Sauer, Dissipative Scattering of Spinning Black Holes at Fourth Post-Minkowskian Order, Phys. Rev. Lett. 131, 241402 (2023), arXiv:2308.11514 [hep-th].arxiv.org/abs/2308.11514
  108. [108]P. H. Damgaard, E. R. Hansen, L. Planté, and P. Vanhove, Classical observables from the exponential representation of the gravitational S-matrix, JHEP 09, 183, arXiv:2307.04746 [hep-th].arxiv.org/abs/2307.04746
  109. [109]E. Herrmann, J. Parra-Martinez, M. S. Ruf, and M. Zeng, Gravitational Bremsstrahlung from Reverse Unitarity, Phys. Rev. Lett. 126, 201602 (2021), arXiv:2101.07255 [hep-th].DOI
  110. [110]M. M. Riva, F. Vernizzi, and L. K. Wong, Angular momentum balance in gravitational two-body scattering: Flux, memory, and supertranslation invariance, Phys. Rev. D 108, 104052 (2023), arXiv:2302.09065 [gr-qc].arxiv.org/abs/2302.09065
  111. [111]G. U. Jakobsen and K. Stoldt, Runaway to dissipation of angular momentum via worldline quantum field theory, Phys. Rev. D 114, 064020 (2026), arXiv:2605.12214 [hep-th].arxiv.org/abs/2605.12214
  112. [112]C. Heissenberg, Radiation-reaction and angular momentum loss at O(G^4), Phys. Rev. D 111, 126012 (2025), arXiv:2501.02904 [hep-th].arxiv.org/abs/2501.02904
  113. [113]A. Georgoudis, C. Heissenberg, G. Mogull, and R. Russo, Angular Momentum Loss at Fourth Post-Minkowskian Order (2026), arXiv:2610.24072 [hep-th].arxiv.org/abs/2610.24072
  114. [114]L. Bohnenblust, C. Jordan Eriksen, G. U. Jakobsen, K. Stoldt, and A. Yanovets, Mechanical angular momentum loss at O(G^4) from static and radiative gravitons (2026), arXiv:2610.26822 [hep-th].arxiv.org/abs/2610.26822
  115. [115]M. Driesse, G. U. Jakobsen, G. Mogull, J. Plefka, B. Sauer, and J. Usovitsch, Conservative Black Hole Scattering at Fifth Post-Minkowskian and First Self-Force Order, Phys. Rev. Lett. 132, 241402 (2024), arXiv:2403.07781 [hep-th].arxiv.org/abs/2403.07781
  116. [116]Z. Bern, E. Herrmann, R. Roiban, M. S. Ruf, A. V. Smirnov, V. A. Smirnov, and M. Zeng, Amplitudes, supersymmetric black hole scattering at O (G^5), and loop integration, JHEP 10, 023, arXiv:2406.01554 [hep-th].arxiv.org/abs/2406.01554
  117. [117]M. Driesse, G. U. Jakobsen, A. Klemm, G. Mogull, C. Nega, J. Plefka, B. Sauer, and J. Usovitsch, Emergence of Calabi–Yau manifolds in high-precision black-hole scattering, Nature 641, 603 (2025), arXiv:2411.11846 [hep-th].DOI
  118. [118]C. Dlapa, G. Kälin, Z. Liu, and R. A. Porto, Local-in-Time Conservative Binary Dynamics at Fifth Post-Minkowskian and First Self-Force Orders, Phys. Rev. Lett. 135, 251401 (2025), arXiv:2506.20665 [hep-th].arxiv.org/abs/2506.20665
  119. [119]M. Driesse, G. U. Jakobsen, G. Mogull, C. Nega, J. Plefka, B. Sauer, and J. Usovitsch, Conservative Black Hole Scattering at Fifth Post-Minkowskian and Second Self-Force Order, Phys. Rev. Lett. 137, 081402 (2026), arXiv:2601.16256 [hep-th].arxiv.org/abs/2601.16256
  120. [120]Z. Bern, E. Herrmann, R. Roiban, M. S. Ruf, A. V. Smirnov, S. Smith, and M. Zeng, Scattering Amplitudes and Conservative Binary Dynamics at O(G^5) without Self-Force Truncation, Phys. Rev. Lett. 137, 021402 (2026), arXiv:2512.23654 [hep-th].arxiv.org/abs/2512.23654
  121. [121]C. Dlapa, G. Kälin, Z. Liu, and R. A. Porto, Nonlocal-in-time tail effects in gravitational scattering to fifth post-Minkowskian and tenth self-force orders, Phys. Rev. D 114, 024029 (2026), arXiv:2604.25916 [hep-th].arxiv.org/abs/2604.25916
  122. [122]W. L. Burke, Gravitational radiation damping of slowly moving systems calculated using matched asymptotic expansions, J. Math. Phys. 12, 401 (1971).DOI
  123. [123]B. R. Iyer and C. M. Will, Post-Newtonian gravitational radiation reaction for two-body systems, Phys. Rev. Lett. 70, 113 (1993).
  124. [124]B. R. Iyer and C. M. Will, Post-Newtonian gravitational radiation reaction for two-body systems: Nonspinning bodies, Phys. Rev. D 52, 6882 (1995).
  125. [125]C. R. Galley and M. Tiglio, Radiation reaction and gravitational waves in the effective field theory approach, Phys. Rev. D 79, 124027 (2009), arXiv:0903.1122 [gr-qc].DOI
  126. [126]C. R. Galley and A. K. Leibovich, Radiation reaction at 3.5 post-Newtonian order in effective field theory, Phys. Rev. D 86, 044029 (2012), arXiv:1205.3842 [gr-qc].DOI
  127. [127]A. K. Leibovich, N. T. Maia, I. Z. Rothstein, and Z. Yang, Second post-Newtonian order radiative dynamics of inspiralling compact binaries in the Effective Field Theory approach, Phys. Rev. D 101, 084058 (2020), arXiv:1912.12546 [gr-qc].DOI
  128. [128]A. K. Leibovich, B. A. Pardo, and Z. Yang, Radiation reaction for nonspinning bodies at 4.5PN in the effective field theory approach, Phys. Rev. D 108, 024017 (2023), arXiv:2302.11016 [gr-qc].DOI
  129. [129]L. Blanchet and T. Damour, Tail Transported Temporal Correlations in the Dynamics of a Gravitating System, Phys. Rev. D 37, 1410 (1988).pubmed.ncbi.nlm.nih.gov/9958830
  130. [130]L. Blanchet and T. Damour, Hereditary effects in gravitational radiation, Phys. Rev. D 46, 4304 (1992).pubmed.ncbi.nlm.nih.gov/10014811
  131. [131]L. Blanchet, Gravitational wave tails of tails, Class. Quant. Grav. 15, 113 (1998), [Erratum: Class.Quant.Grav. 22, 3381 (2005)], arXiv:gr-qc/9710038.DOI
  132. [132]G. F. L. Almeida, A. Müller, S. Foffa, and R. Sturani, Conservative binary dynamics from gravitational tail emission processes, Phys. Rev. D 108, 124010 (2023), [Erratum: Phys.Rev.D 113, 089903 (2026)], arXiv:2307.05327 [gr-qc].arxiv.org/abs/2307.05327
  133. [133]Q. Henry and F. Larroutourou, Conservative tail and failed-tail effects at the fifth post-Newtonian order, Phys. Rev. D 108, 084048 (2023), arXiv:2307.05860 [gr-qc].DOI
  134. [134]G. L. Almeida, A. Müller, S. Foffa, and R. Sturani, Angular momentum tail contributions to compact binary dynamics, Phys. Rev. D 114, 064057 (2026), arXiv:2603.20096 [gr-qc].DOI
  135. [135]D. Christodoulou, Nonlinear nature of gravitation and gravitational wave experiments, Phys. Rev. Lett. 67, 1486 (1991).pubmed.ncbi.nlm.nih.gov/10044168
  136. [136]A. G. Wiseman and C. M. Will, Christodoulou’s non-linear gravitational wave memory: Evaluation in the quadrupole approximation, Phys. Rev. D 44, R2945 (1991).
  137. [137]K. S. Thorne, Gravitational-wave bursts with memory: The Christodoulou effect, Phys. Rev. D 45, 520 (1992).pubmed.ncbi.nlm.nih.gov/10014404
  138. [138]M. Favata, Post-Newtonian corrections to the gravitational-wave memory for quasi-circular, inspiralling compact binaries, Phys. Rev. D 80, 024002 (2009), arXiv:0812.0069 [gr-qc].DOI
  139. [139]D. Trestini and L. Blanchet, Gravitational-wave tails of memory, Phys. Rev. D 107, 104048 (2023), arXiv:2301.09395 [gr-qc].DOI
  140. [140]J. S. Schwinger, Brownian motion of a quantum oscillator, J. Math. Phys. 2, 407 (1961).
  141. [141]L. V. Keldysh, Diagram Technique for Nonequilibrium Processes, Sov. Phys. JETP 20, 1018 (1965).
  142. [142]E. Calzetta and B. L. Hu, Closed Time Path Functional Formalism in Curved Space-Time: Application to Cosmological Back Reaction Problems, Phys. Rev. D 35, 495 (1987).pubmed.ncbi.nlm.nih.gov/9957683
  143. [143]C. R. Galley, Classical Mechanics of Nonconservative Systems, Phys. Rev. Lett. 110, 174301 (2013), arXiv:1210.2745 [gr-qc].
  144. [144]C. R. Galley, D. Tsang, and L. C. Stein, The principle of stationary nonconservative action for classical mechanics and field theories (2014), arXiv:1412.3082 [math-ph].arxiv.org/abs/1412.3082
  145. [145]S. Foffa and R. Sturani, Tail terms in gravitational radiation reaction via effective field theory, Phys. Rev. D 87, 044056 (2013), arXiv:1111.5488 [gr-qc].DOI
  146. [146]S. Foffa and R. Sturani, Hereditary terms at next-to-leading order in two-body gravitational dynamics, Phys. Rev. D 101, 064033 (2020), [Erratum: Phys.Rev.D 103, 089901 (2021)], arXiv:1907.02869 [gr-qc].DOI
  147. [147]G. Brunello, Effective Field Theory Approach to General Relativity and Feynman Diagrams for Coalescing Binary Systems, Master’s thesis, U. Padua, Dept. Phys. Astron. (2022), arXiv:2211.01321 [hep-th].arxiv.org/abs/2211.01321
  148. [148]G. L. Almeida, S. Foffa, and R. Sturani, Gravitational radiation contributions to the two-body scattering angle, Phys. Rev. D 107, 024020 (2023), arXiv:2209.11594 [gr-qc].DOI
  149. [149]G. L. Almeida, A. Müller, S. Foffa, and R. Sturani, Gravitational memory contributions to waveform and effective action, Phys. Rev. D 112, 084077 (2025), arXiv:2410.10565 [gr-qc].arxiv.org/abs/2410.10565
  150. [150]G. L. Almeida, S. Foffa, and R. Sturani, Tail contributions to gravitational conservative dynamics, Phys. Rev. D 104, 124075 (2021), arXiv:2110.14146 [gr-qc].DOI
  151. [151]Q. Henry, G. Faye, and L. Blanchet, The current-type quadrupole moment and gravitational-wave mode (ℓ, m) = (2, 1) of compact binary systems at the third post-Newtonian order, Class. Quant. Grav. 38, 185004 (2021), arXiv:2105.10876 [gr-qc].DOI
  152. [152]F. Larroutourou, L. Blanchet, Q. Henry, and G. Faye, The quadrupole moment of compact binaries to the fourth post-Newtonian order: II. Dimensional regularization and renormalization, Class. Quant. Grav. 39, 115008 (2022), arXiv:2110.02243 [gr-qc].
  153. [153]D. Trestini, F. Larroutourou, and L. Blanchet, The quadrupole moment of compact binaries to the fourth post-Newtonian order: relating the harmonic and radiative metrics, Class. Quant. Grav. 40, 055006 (2023), arXiv:2209.02719 [gr-qc].DOI
  154. [154]M. Beneke and V. A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B 522, 321 (1998), arXiv:hep-ph/9711391.DOI
  155. [155]D. Bini, T. Damour, and G. Faye, Effective action approach to higher-order relativistic tidal interactions in binary systems and their effective one body description, Phys. Rev. D 85, 124034 (2012), arXiv:1203.2565 [gr-qc].
  156. [156]R. A. Porto, The Tune of Love and the Nature(ness) of Spacetime, Fortsch. Phys. 64, 723 (2016), arXiv:1606.08895 [gr-qc].DOI
  157. [157]L. Bohnenblust, C. Jordan Eriksen, J. Høgeveen, G. U. Jakobsen, and J. Plefka, Black hole response theory and its exact shockwave limit, JHEP 08, 064, arXiv:2604.22009 [hep-th].
  158. [158]F. A. Berends and W. T. Giele, Recursive Calculations for Processes with n Gluons, Nucl. Phys. B 306, 759 (1988).DOI
  159. [159]G. U. Jakobsen, G. Mogull, J. Plefka, B. Sauer, and Y. Xu, Conservative Scattering of Spinning Black Holes at Fourth Post-Minkowskian Order, Phys. Rev. Lett. 131, 151401 (2023), arXiv:2306.01714 [hep-th].arxiv.org/abs/2306.01714
  160. [160]Y. F. Bautista, M. Driesse, K. Haddad, and G. U. Jakobsen, Gravitational wave scattering in spinless WQFT, JHEP 05, 252, arXiv:2602.06125 [hep-th].
  161. [161]G. Brunello, M. Meo, and S. Smith, Gravitational Compton scattering at the fourth post-Minkowskian order, Phys. Rev. D 114, 085022 (2026), arXiv:2606.28239 [hep-th].DOI
  162. [162]G. Kälin and J. Lim, Graph-Based Algorithms for Worldline EFT Integrands (2026), arXiv:2610.35007 [hep-th].
  163. [163]T. Marchand, L. Blanchet, and G. Faye, Gravitational-wave tail effects to quartic non-linear order, Class. Quant. Grav. 33, 244003 (2016), arXiv:1607.07601 [gr-qc].
  164. [164]A. Edison and M. Levi, Higher-order tails and RG flows due to scattering of gravitational radiation from binary inspirals, JHEP 08, 161, arXiv:2310.20066 [hep-th].
  165. [165]J. A. M. Vermaseren, New features of FORM (2000), arXiv:math-ph/0010025.DOI
  166. [166]B. Ruijl, T. Ueda, and J. Vermaseren, FORM version 4.2 (2017), arXiv:1707.06453 [hep-ph].arxiv.org/abs/1707.06453
  167. [167]K. G. Chetyrkin and F. V. Tkachov, Integration by parts: The algorithm to calculate β-functions in 4 loops, Nucl. Phys. B 192, 159 (1981).DOI
  168. [168]S. Laporta, High-precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15, 5087 (2000), arXiv:hep-ph/0102033.DOI
  169. [169]R. N. Lee, Presenting LiteRed: a tool for the Loop InTEGrals REDuction (2012), arXiv:1212.2685 [hep-ph].arxiv.org/abs/1212.2685
  170. [170]R. N. Lee, LiteRed 1.4: a powerful tool for reduction of multiloop integrals, J. Phys. Conf. Ser. 523, 012059 (2014), arXiv:1310.1145 [hep-ph].DOI
  171. [171]L. Blanchet, The asymmetric structure of gravitational radiation, Phys. Rev. D 47, 4392 (1993).pubmed.ncbi.nlm.nih.gov/10015438
  172. [172]K. S. Thorne, Multipole Expansions of Gravitational Radiation, Rev. Mod. Phys. 52, 299 (1980).DOI
  173. [173]H. Poincaré, Leçons de mécanique céleste, Vol. 3 (Gauthier-Villars, Paris, 1910).
  174. [174]G. Bertrand, La théorie des marées et les équations intégrales, Ann. Sci. Éc. Norm. Supér. 40, 151 (1923).
  175. [175]N. I. Muskhelishvili, Singular Integral Equations (P. Noordhoff, Groningen, 1953).
  176. [176]P. Di Vecchia, C. Heissenberg, and R. Russo, Angular momentum of zero-frequency gravitons, JHEP 08, 172, arXiv:2203.11915 [hep-th].DOI
  177. [177]D. Bini, T. Damour, and A. Geralico, Radiated momentum and radiation reaction in gravitational two-body scattering including time-asymmetric effects, Phys. Rev. D 107, 024012 (2023), arXiv:2210.07165 [gr-qc].DOI
  178. [178]G. Faye, L. Blanchet, and B. R. Iyer, Non-linear multipole interactions and gravitational-wave octupole modes for inspiralling compact binaries to third-and-a-half post-Newtonian order, Class. Quant. Grav. 32, 045016 (2015), arXiv:1409.3546 [gr-qc].DOI
  179. [179]G. Kălin and R. A. Porto, From boundary data to bound states. Part II. Scattering angle to dynamical invariants (with twist), JHEP 02, 120, arXiv:1911.09130 [hep-th].
  180. [180]T. Damour, Classical and quantum scattering in post-Minkowskian gravity, Phys. Rev. D 102, 024060 (2020), arXiv:1912.02139 [gr-qc].DOI
  181. [181]T. Damour and B. R. Iyer, Multipole analysis for electromagnetism and linearized gravity with irreducible cartesian tensors, Phys. Rev. D 43, 3259 (1991).pubmed.ncbi.nlm.nih.gov/10013280

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