Abstract
We study the classical dynamics of spinning particles using scattering amplitudes and eikonal exponentiation. We show that observables are determined by a simple algorithm. A wealth of complexity arises in perturbation theory as positions, momenta and spins must be iteratively corrected at each order. Even though we restrict ourselves to one-loop computations at quadratic order in spin, nevertheless we encounter and resolve a number of subtle effects. Finally, we clarify the links between our work and various other eikonal approaches to spinning observables.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
LIGO Scientific and Virgo collaborations, Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett. 116 (2016) 061102 [arXiv:1602.03837] [INSPIRE].
LIGO Scientific and Virgo collaborations, GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral, Phys. Rev. Lett. 119 (2017) 161101 [arXiv:1710.05832] [INSPIRE].
T. Damour, Gravitational scattering, post-Minkowskian approximation and Effective One-Body theory, Phys. Rev. D 94 (2016) 104015 [arXiv:1609.00354] [INSPIRE].
T. Damour, High-energy gravitational scattering and the general relativistic two-body problem, Phys. Rev. D 97 (2018) 044038 [arXiv:1710.10599] [INSPIRE].
N.E.J. Bjerrum-Bohr et al., General Relativity from Scattering Amplitudes, Phys. Rev. Lett. 121 (2018) 171601 [arXiv:1806.04920] [INSPIRE].
C. Cheung, I.Z. Rothstein and M.P. Solon, From Scattering Amplitudes to Classical Potentials in the Post-Minkowskian Expansion, Phys. Rev. Lett. 121 (2018) 251101 [arXiv:1808.02489] [INSPIRE].
D.A. Kosower, B. Maybee and D. O’Connell, Amplitudes, Observables, and Classical Scattering, JHEP 02 (2019) 137 [arXiv:1811.10950] [INSPIRE].
A. Cristofoli, N.E.J. Bjerrum-Bohr, P.H. Damgaard and P. Vanhove, Post-Minkowskian Hamiltonians in general relativity, Phys. Rev. D 100 (2019) 084040 [arXiv:1906.01579] [INSPIRE].
N.E.J. Bjerrum-Bohr, A. Cristofoli and P.H. Damgaard, Post-Minkowskian Scattering Angle in Einstein Gravity, JHEP 08 (2020) 038 [arXiv:1910.09366] [INSPIRE].
A. Brandhuber, G. Chen, G. Travaglini and C. Wen, Classical gravitational scattering from a gauge-invariant double copy, JHEP 10 (2021) 118 [arXiv:2108.04216] [INSPIRE].
Z. Bern et al., The Duality Between Color and Kinematics and its Applications, arXiv:1909.01358 [INSPIRE].
J. Parra-Martinez, M.S. Ruf and M. Zeng, Extremal black hole scattering at \( \mathcal{O} \)(G3): graviton dominance, eikonal exponentiation, and differential equations, JHEP 11 (2020) 023 [arXiv:2005.04236] [INSPIRE].
Z. Bern et al., Scattering Amplitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order, Phys. Rev. Lett. 122 (2019) 201603 [arXiv:1901.04424] [INSPIRE].
Z. Bern et al., Black Hole Binary Dynamics from the Double Copy and Effective Theory, JHEP 10 (2019) 206 [arXiv:1908.01493] [INSPIRE].
Z. Bern et al., Scattering Amplitudes and Conservative Binary Dynamics at \( \mathcal{O} \)(G4), Phys. Rev. Lett. 126 (2021) 171601 [arXiv:2101.07254] [INSPIRE].
Z. Bern et al., Scattering Amplitudes, the Tail Effect, and Conservative Binary Dynamics at O(G4), Phys. Rev. Lett. 128 (2022) 161103 [arXiv:2112.10750] [INSPIRE].
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 (2020) 261103 [arXiv:2007.04977] [INSPIRE].
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 (2022) 137203 [arXiv:2106.08276] [INSPIRE].
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 (2022) 161104 [arXiv:2112.11296] [INSPIRE].
C. Dlapa et al., Radiation Reaction and Gravitational Waves at Fourth Post-Minkowskian Order, Phys. Rev. Lett. 130 (2023) 101401 [arXiv:2210.05541] [INSPIRE].
N.E.J. Bjerrum-Bohr, L. Planté and P. Vanhove, Effective Field Theory and Applications: Weak Field Observables from Scattering Amplitudes in Quantum Field Theory, arXiv:2212.08957 [INSPIRE].
G.U. Jakobsen et al., Conservative Scattering of Spinning Black Holes at Fourth Post-Minkowskian Order, Phys. Rev. Lett. 131 (2023) 151401 [arXiv:2306.01714] [INSPIRE].
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 (2023) 241402 [arXiv:2308.11514] [INSPIRE].
M. Punturo et al., The Einstein Telescope: A third-generation gravitational wave observatory, Class. Quant. Grav. 27 (2010) 194002 [INSPIRE].
LISA collaboration, Laser Interferometer Space Antenna, arXiv:1702.00786 [INSPIRE].
D. Reitze et al., Cosmic Explorer: The U.S. Contribution to Gravitational-Wave Astronomy beyond LIGO, Bull. Am. Astron. Soc. 51 (2019) 035 [arXiv:1907.04833] [INSPIRE].
D. Bini and T. Damour, Gravitational spin-orbit coupling in binary systems, post-Minkowskian approximation and effective one-body theory, Phys. Rev. D 96 (2017) 104038 [arXiv:1709.00590] [INSPIRE].
J. Vines, Scattering of two spinning black holes in post-Minkowskian gravity, to all orders in spin, and effective-one-body mappings, Class. Quant. Grav. 35 (2018) 084002 [arXiv:1709.06016] [INSPIRE].
D. Bini and T. Damour, Gravitational spin-orbit coupling in binary systems at the second post-Minkowskian approximation, Phys. Rev. D 98 (2018) 044036 [arXiv:1805.10809] [INSPIRE].
P.H. Damgaard, J. Hoogeveen, A. Luna and J. Vines, Scattering angles in Kerr metrics, Phys. Rev. D 106 (2022) 124030 [arXiv:2208.11028] [INSPIRE].
J. Hoogeveen, Charged test particle scattering and effective one-body metrics with spin, Phys. Rev. D 108 (2023) 024049 [arXiv:2303.00317] [INSPIRE].
A. Guevara, Holomorphic Classical Limit for Spin Effects in Gravitational and Electromagnetic Scattering, JHEP 04 (2019) 033 [arXiv:1706.02314] [INSPIRE].
A. Guevara, A. Ochirov and J. Vines, Scattering of Spinning Black Holes from Exponentiated Soft Factors, JHEP 09 (2019) 056 [arXiv:1812.06895] [INSPIRE].
M.-Z. Chung, Y.-T. Huang, J.-W. Kim and S. Lee, The simplest massive S-matrix: from minimal coupling to Black Holes, JHEP 04 (2019) 156 [arXiv:1812.08752] [INSPIRE].
N. Moynihan, Kerr-Newman from Minimal Coupling, JHEP 01 (2020) 014 [arXiv:1909.05217] [INSPIRE].
M.-Z. Chung, Y.-T. Huang and J.-W. Kim, Kerr-Newman stress-tensor from minimal coupling, JHEP 12 (2020) 103 [arXiv:1911.12775] [INSPIRE].
M.-Z. Chung, Y.-T. Huang and J.-W. Kim, Classical potential for general spinning bodies, JHEP 09 (2020) 074 [arXiv:1908.08463] [INSPIRE].
A. Guevara, A. Ochirov and J. Vines, Black-hole scattering with general spin directions from minimal-coupling amplitudes, Phys. Rev. D 100 (2019) 104024 [arXiv:1906.10071] [INSPIRE].
M.-Z. Chung, Y.-T. Huang, J.-W. Kim and S. Lee, Complete Hamiltonian for spinning binary systems at first post-Minkowskian order, JHEP 05 (2020) 105 [arXiv:2003.06600] [INSPIRE].
J. Vines, J. Steinhoff and A. Buonanno, Spinning-black-hole scattering and the test-black-hole limit at second post-Minkowskian order, Phys. Rev. D 99 (2019) 064054 [arXiv:1812.00956] [INSPIRE].
M. Chiodaroli, H. Johansson and P. Pichini, Compton black-hole scattering for s ≤ 5/2, JHEP 02 (2022) 156 [arXiv:2107.14779] [INSPIRE].
L. Cangemi and P. Pichini, Classical limit of higher-spin string amplitudes, JHEP 06 (2023) 167 [arXiv:2207.03947] [INSPIRE].
L. Cangemi et al., Kerr Black Holes From Massive Higher-Spin Gauge Symmetry, Phys. Rev. Lett. 131 (2023) 221401 [arXiv:2212.06120] [INSPIRE].
A. Ochirov and E. Skvortsov, Chiral Approach to Massive Higher Spins, Phys. Rev. Lett. 129 (2022) 241601 [arXiv:2207.14597] [INSPIRE].
L. Cangemi et al., From higher-spin gauge interactions to Compton amplitudes for root-Kerr, arXiv:2311.14668 [INSPIRE].
F. Alessio, Kerr binary dynamics from minimal coupling and double copy, JHEP 04 (2024) 058 [arXiv:2303.12784] [INSPIRE].
Y.F. Bautista, A. Guevara, C. Kavanagh and J. Vines, Scattering in black hole backgrounds and higher-spin amplitudes. Part II, JHEP 05 (2023) 211 [arXiv:2212.07965] [INSPIRE].
Y.F. Bautista et al., Black hole perturbation theory meets CFT2: Kerr-Compton amplitudes from Nekrasov-Shatashvili functions, Phys. Rev. D 109 (2024) 084071 [arXiv:2312.05965] [INSPIRE].
Y.F. Bautista, Dynamics for super-extremal Kerr binary systems at \( \mathcal{O} \)(G2), Phys. Rev. D 108 (2023) 084036 [arXiv:2304.04287] [INSPIRE].
Z. Bern et al., Spinning black hole binary dynamics, scattering amplitudes, and effective field theory, Phys. Rev. D 104 (2021) 065014 [arXiv:2005.03071] [INSPIRE].
Z. Bern et al., Leading Nonlinear Tidal Effects and Scattering Amplitudes, JHEP 05 (2021) 188 [arXiv:2010.08559] [INSPIRE].
D. Kosmopoulos and A. Luna, Quadratic-in-spin Hamiltonian at \( \mathcal{O} \)(G2) from scattering amplitudes, JHEP 07 (2021) 037 [arXiv:2102.10137] [INSPIRE].
F. Febres Cordero et al., Conservative Binary Dynamics with a Spinning Black Hole at \( \mathcal{O} \)(G3) from Scattering Amplitudes, Phys. Rev. Lett. 130 (2023) 021601 [arXiv:2205.07357] [INSPIRE].
Z. Bern et al., Binary Dynamics through the Fifth Power of Spin at \( \mathcal{O} \)(G2), Phys. Rev. Lett. 130 (2023) 201402 [arXiv:2203.06202] [INSPIRE].
B. Maybee, D. O’Connell and J. Vines, Observables and amplitudes for spinning particles and black holes, JHEP 12 (2019) 156 [arXiv:1906.09260] [INSPIRE].
G. Menezes and M. Sergola, NLO deflections for spinning particles and Kerr black holes, JHEP 10 (2022) 105 [arXiv:2205.11701] [INSPIRE].
P.H. Damgaard, K. Haddad and A. Helset, Heavy Black Hole Effective Theory, JHEP 11 (2019) 070 [arXiv:1908.10308] [INSPIRE].
R. Aoude, K. Haddad and A. Helset, On-shell heavy particle effective theories, JHEP 05 (2020) 051 [arXiv:2001.09164] [INSPIRE].
R. Aoude, K. Haddad and A. Helset, Tidal effects for spinning particles, JHEP 03 (2021) 097 [arXiv:2012.05256] [INSPIRE].
R. Aoude and A. Ochirov, Classical observables from coherent-spin amplitudes, JHEP 10 (2021) 008 [arXiv:2108.01649] [INSPIRE].
W.-M. Chen, M.-Z. Chung, Y.-T. Huang and J.-W. Kim, The 2PM Hamiltonian for binary Kerr to quartic in spin, JHEP 08 (2022) 148 [arXiv:2111.13639] [INSPIRE].
W.-M. Chen, M.-Z. Chung, Y.-T. Huang and J.-W. Kim, Gravitational Faraday effect from on-shell amplitudes, JHEP 12 (2022) 058 [arXiv:2205.07305] [INSPIRE].
R. Aoude, K. Haddad and A. Helset, Searching for Kerr in the 2PM amplitude, JHEP 07 (2022) 072 [arXiv:2203.06197] [INSPIRE].
N.E.J. Bjerrum-Bohr, G. Chen and M. Skowronek, Classical spin gravitational Compton scattering, JHEP 06 (2023) 170 [arXiv:2302.00498] [INSPIRE].
K. Haddad, Recursion in the classical limit and the neutron-star Compton amplitude, JHEP 05 (2023) 177 [arXiv:2303.02624] [INSPIRE].
R. Aoude, K. Haddad and A. Helset, Classical gravitational scattering amplitude at O(G2S1∞S2∞), Phys. Rev. D 108 (2023) 024050 [arXiv:2304.13740] [INSPIRE].
N.E.J. Bjerrum-Bohr, G. Chen and M. Skowronek, Covariant Compton Amplitudes in Gravity with Classical Spin, Phys. Rev. Lett. 132 (2024) 191603 [arXiv:2309.11249] [INSPIRE].
A. Brandhuber et al., Resummed spinning waveforms from five-point amplitudes, JHEP 02 (2024) 026 [arXiv:2310.04405] [INSPIRE].
S. De Angelis, R. Gonzo and P.P. Novichkov, Spinning waveforms from KMOC at leading order, arXiv:2309.17429 [INSPIRE].
M.V.S. Saketh, J. Steinhoff, J. Vines and A. Buonanno, Modeling horizon absorption in spinning binary black holes using effective worldline theory, Phys. Rev. D 107 (2023) 084006 [arXiv:2212.13095] [INSPIRE].
R. Aoude and A. Ochirov, Gravitational partial-wave absorption from scattering amplitudes, JHEP 12 (2023) 103 [arXiv:2307.07504] [INSPIRE].
R. Aoude, K. Haddad, C. Heissenberg and A. Helset, Leading-order gravitational radiation to all spin orders, Phys. Rev. D 109 (2024) 036007 [arXiv:2310.05832] [INSPIRE].
Y.-J. Chen, T. Hsieh, Y.-T. Huang and J.-W. Kim, On-shell approach to (spinning) gravitational absorption processes, arXiv:2312.04513 [INSPIRE].
M.V.S. Saketh, J. Vines, J. Steinhoff and A. Buonanno, Conservative and radiative dynamics in classical relativistic scattering and bound systems, Phys. Rev. Res. 4 (2022) 013127 [arXiv:2109.05994] [INSPIRE].
Z. Bern et al., Scalar QED as a toy model for higher-order effects in classical gravitational scattering, JHEP 08 (2022) 131 [arXiv:2112.12243] [INSPIRE].
N. Arkani-Hamed, Y.-T. Huang and D. O’Connell, Kerr black holes as elementary particles, JHEP 01 (2020) 046 [arXiv:1906.10100] [INSPIRE].
J.-W. Kim and J. Steinhoff, Spin supplementary condition in quantum field theory: covariant SSC and physical state projection, JHEP 07 (2023) 042 [arXiv:2302.01944] [INSPIRE].
Z. Bern et al., Quantum field theory, worldline theory, and spin magnitude change in orbital evolution, Phys. Rev. D 109 (2024) 045011 [arXiv:2308.14176] [INSPIRE].
L. de la Cruz, B. Maybee, D. O’Connell and A. Ross, Classical Yang-Mills observables from amplitudes, JHEP 12 (2020) 076 [arXiv:2009.03842] [INSPIRE].
L. de la Cruz, A. Luna and T. Scheopner, Yang-Mills observables: from KMOC to eikonal through EFT, JHEP 01 (2022) 045 [arXiv:2108.02178] [INSPIRE].
Z. Liu, R.A. Porto and Z. Yang, Spin Effects in the Effective Field Theory Approach to Post-Minkowskian Conservative Dynamics, JHEP 06 (2021) 012 [arXiv:2102.10059] [INSPIRE].
G.U. Jakobsen, G. Mogull, J. Plefka and J. Steinhoff, Gravitational Bremsstrahlung and Hidden Supersymmetry of Spinning Bodies, Phys. Rev. Lett. 128 (2022) 011101 [arXiv:2106.10256] [INSPIRE].
G.U. Jakobsen, G. Mogull, J. Plefka and J. Steinhoff, SUSY in the sky with gravitons, JHEP 01 (2022) 027 [arXiv:2109.04465] [INSPIRE].
M. Ben-Shahar, Scattering of spinning compact objects from a worldline EFT, JHEP 03 (2024) 108 [arXiv:2311.01430] [INSPIRE].
T. Scheopner and J. Vines, Dynamical Implications of the Kerr Multipole Moments for Spinning Black Holes, arXiv:2311.18421 [INSPIRE].
M. Ciafaloni, D. Colferai and G. Veneziano, Infrared features of gravitational scattering and radiation in the eikonal approach, Phys. Rev. D 99 (2019) 066008 [arXiv:1812.08137] [INSPIRE].
P. Di Vecchia, C. Heissenberg, R. Russo and G. Veneziano, The eikonal approach to gravitational scattering and radiation at \( \mathcal{O} \)(G3), JHEP 07 (2021) 169 [arXiv:2104.03256] [INSPIRE].
A. Cristofoli et al., The uncertainty principle and classical amplitudes, JHEP 06 (2024) 181 [arXiv:2112.07556] [INSPIRE].
C. Heissenberg, Infrared divergences and the eikonal exponentiation, Phys. Rev. D 104 (2021) 046016 [arXiv:2105.04594] [INSPIRE].
W.T. Emond, N. Moynihan and L. Wei, Quantization conditions and the double copy, JHEP 09 (2022) 108 [arXiv:2109.11531] [INSPIRE].
P. Di Vecchia, C. Heissenberg, R. Russo and G. Veneziano, The eikonal operator at arbitrary velocities I: the soft-radiation limit, JHEP 07 (2022) 039 [arXiv:2204.02378] [INSPIRE].
K. Haddad, Exponentiation of the leading eikonal phase with spin, Phys. Rev. D 105 (2022) 026004 [arXiv:2109.04427] [INSPIRE].
T. Adamo, A. Cristofoli and P. Tourkine, Eikonal amplitudes from curved backgrounds, SciPost Phys. 13 (2022) 032 [arXiv:2112.09113] [INSPIRE].
P. Di Vecchia, C. Heissenberg, R. Russo and G. Veneziano, Classical gravitational observables from the Eikonal operator, Phys. Lett. B 843 (2023) 138049 [arXiv:2210.12118] [INSPIRE].
J.P. Gatica, The Eikonal Phase and Spinning Observables, arXiv:2312.04680 [INSPIRE].
A. Georgoudis, C. Heissenberg and R. Russo, An eikonal-inspired approach to the gravitational scattering waveform, JHEP 03 (2024) 089 [arXiv:2312.07452] [INSPIRE].
P. Di Vecchia, C. Heissenberg, R. Russo and G. Veneziano, The gravitational eikonal: From particle, string and brane collisions to black-hole encounters, Phys. Rept. 1083 (2024) 1 [arXiv:2306.16488] [INSPIRE].
C. Heissenberg, Angular momentum loss due to spin-orbit effects in the post-Minkowskian expansion, Phys. Rev. D 108 (2023) 106003 [arXiv:2308.11470] [INSPIRE].
F. Alessio and P. Di Vecchia, Radiation reaction for spinning black-hole scattering, Phys. Lett. B 832 (2022) 137258 [arXiv:2203.13272] [INSPIRE].
M. Bianchi, C. Gambino and F. Riccioni, A Rutherford-like formula for scattering off Kerr-Newman BHs and subleading corrections, JHEP 08 (2023) 188 [arXiv:2306.08969] [INSPIRE].
G.U. Jakobsen and G. Mogull, Linear response, Hamiltonian, and radiative spinning two-body dynamics, Phys. Rev. D 107 (2023) 044033 [arXiv:2210.06451] [INSPIRE].
G. Kälin and R.A. Porto, Post-Minkowskian Effective Field Theory for Conservative Binary Dynamics, JHEP 11 (2020) 106 [arXiv:2006.01184] [INSPIRE].
A. Cristofoli, R. Gonzo, D.A. Kosower and D. O’Connell, Waveforms from amplitudes, Phys. Rev. D 106 (2022) 056007 [arXiv:2107.10193] [INSPIRE].
Acknowledgments
We thank Giulia Isabella, Guanda Lin, Jan Steinhoff, Fei Teng, Justin Vines and Mao Zeng for useful discussions. A. L. is supported by funds from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No. 847523 ‘INTERACTIONS’. The work of NM was supported by the Science and Technology Facilities Council (STFC) Consolidated Grants ST/T000686/1 “Amplitudes, Strings & Duality” and ST/X00063X/1 “Amplitudes, Strings & Duality”. No new data were generated or analysed during this study. DOC is supported in part by the STFC grant “Particle Physics at the Higgs Centre”. We are also grateful to the organisers of the “Amplifying gravity at all scales” program at Nordita, where parts of this project were undertaken.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2312.09960
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Luna, A., Moynihan, N., O’Connell, D. et al. Observables from the spinning eikonal. J. High Energ. Phys. 2024, 45 (2024). https://doi.org/10.1007/JHEP08(2024)045
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2024)045