Abstract
An exponential representation of the S-matrix provides a natural framework for understanding the semi-classical limit of scattering amplitudes. While sharing some similarities with the eikonal formalism it differs from it in details. Computationally, rules are simple because pieces that must be subtracted are given by combinations of unitarity cuts. Analyzing classical gravitational scattering to third Post-Minkowskian order in both maximal supergravity and Einstein gravity we find agreement with other approaches, including the contributions from radiation reaction terms. The kinematical relation for the two-body problem in isotropic coordinates follows immediately from this procedure, again with the inclusion of radiation reaction pieces up to third Post-Minkowskian order.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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, P.H. Damgaard, G. Festuccia, L. Planté and P. Vanhove, 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].
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 (2019) 201603 [arXiv:1901.04424] [INSPIRE].
A. Antonelli, A. Buonanno, J. Steinhoff, M. van de Meent and J. Vines, Energetics of two-body Hamiltonians in post-Minkowskian gravity, Phys. Rev. D 99 (2019) 104004 [arXiv:1901.07102] [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].
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 (2019) 206 [arXiv:1908.01493] [INSPIRE].
G. Kälin and R.A. Porto, From Boundary Data to Bound States, JHEP 01 (2020) 072 [arXiv:1910.03008] [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. Cristofoli, P.H. Damgaard, P. Di Vecchia and C. Heissenberg, Second-order Post-Minkowskian scattering in arbitrary dimensions, JHEP 07 (2020) 122 [arXiv:2003.10274] [INSPIRE].
J. Parra-Martínez, 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].
P. Di Vecchia, C. Heissenberg, R. Russo and G. Veneziano, Universality of ultra-relativistic gravitational scattering, Phys. Lett. B 811 (2020) 135924 [arXiv:2008.12743] [INSPIRE].
T. Damour, Radiative contribution to classical gravitational scattering at the third order in G, Phys. Rev. D 102 (2020) 124008 [arXiv:2010.01641] [INSPIRE].
P. Di Vecchia, C. Heissenberg, R. Russo and G. Veneziano, Radiation Reaction from Soft Theorems, Phys. Lett. B 818 (2021) 136379 [arXiv:2101.05772] [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].
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].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, L. Planté and P. Vanhove, Classical gravity from loop amplitudes, Phys. Rev. D 104 (2021) 026009 [arXiv:2104.04510] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, L. Planté and P. Vanhove, The Amplitude for Classical Gravitational Scattering at Third Post-Minkowskian Order, arXiv:2105.05218 [INSPIRE].
D. Bini, T. Damour and A. Geralico, Radiative contributions to gravitational scattering, Phys. Rev. D 104 (2021) 084031 [arXiv:2107.08896] [INSPIRE].
Y.F. Bautista, A. Guevara, C. Kavanagh and J. Vines, From Scattering in Black Hole Backgrounds to Higher-Spin Amplitudes. Part I, arXiv:2107.10179 [INSPIRE].
A. Cristofoli, R. Gonzo, D.A. Kosower and D. O’Connell, Waveforms from Amplitudes, arXiv:2107.10193 [INSPIRE].
E. Herrmann, J. Parra-Martínez, M.S. Ruf and M. Zeng, Gravitational Bremsstrahlung from Reverse Unitarity, Phys. Rev. Lett. 126 (2021) 201602 [arXiv:2101.07255] [INSPIRE].
E. Herrmann, J. Parra-Martínez, M.S. Ruf and M. Zeng, Radiative classical gravitational observables at \( \mathcal{O} \)(G3) from scattering amplitudes, JHEP 10 (2021) 148 [arXiv:2104.03957] [INSPIRE].
S. Mougiakakos, M.M. Riva and F. Vernizzi, Gravitational Bremsstrahlung in the post-Minkowskian effective field theory, Phys. Rev. D 104 (2021) 024041 [arXiv:2102.08339] [INSPIRE].
G.U. Jakobsen, G. Mogull, J. Plefka and J. Steinhoff, Classical Gravitational Bremsstrahlung from a Worldline Quantum Field Theory, Phys. Rev. Lett. 126 (2021) 201103 [arXiv:2101.12688] [INSPIRE].
G. ’t Hooft, Graviton Dominance in Ultrahigh-Energy Scattering, Phys. Lett. B 198 (1987) 61 [INSPIRE].
D. Amati, M. Ciafaloni and G. Veneziano, Superstring Collisions at Planckian Energies, Phys. Lett. B 197 (1987) 81 [INSPIRE].
D. Amati, M. Ciafaloni and G. Veneziano, Classical and Quantum Gravity Effects from Planckian Energy Superstring Collisions, Int. J. Mod. Phys. A 3 (1988) 1615 [INSPIRE].
D. Amati, M. Ciafaloni and G. Veneziano, Higher Order Gravitational Deflection and Soft Bremsstrahlung in Planckian Energy Superstring Collisions, Nucl. Phys. B 347 (1990) 550 [INSPIRE].
D. Amati, M. Ciafaloni and G. Veneziano, Planckian scattering beyond the semiclassical approximation, Phys. Lett. B 289 (1992) 87 [INSPIRE].
D. Amati, M. Ciafaloni and G. Veneziano, Effective action and all order gravitational eikonal at Planckian energies, Nucl. Phys. B 403 (1993) 707 [INSPIRE].
D.N. Kabat and M. Ortiz, Eikonal quantum gravity and Planckian scattering, Nucl. Phys. B 388 (1992) 570 [hep-th/9203082] [INSPIRE].
R. Akhoury, R. Saotome and G. Sterman, High Energy Scattering in Perturbative Quantum Gravity at Next to Leading Power, Phys. Rev. D 103 (2021) 064036 [arXiv:1308.5204] [INSPIRE].
A. Luna, S. Melville, S.G. Naculich and C.D. White, Next-to-soft corrections to high energy scattering in QCD and gravity, JHEP 01 (2017) 052 [arXiv:1611.02172] [INSPIRE].
A. Koemans Collado, P. Di Vecchia, R. Russo and S. Thomas, The subleading eikonal in supergravity theories, JHEP 10 (2018) 038 [arXiv:1807.04588] [INSPIRE].
A. Koemans Collado, P. Di Vecchia and R. Russo, Revisiting the second post-Minkowskian eikonal and the dynamics of binary black holes, Phys. Rev. D 100 (2019) 066028 [arXiv:1904.02667] [INSPIRE].
G. D’Appollonio, P. Di Vecchia, R. Russo and G. Veneziano, High-energy string-brane scattering: Leading eikonal and beyond, JHEP 11 (2010) 100 [arXiv:1008.4773] [INSPIRE].
G. D’Appollonio, P. Di Vecchia, R. Russo and G. Veneziano, Microscopic unitary description of tidal excitations in high-energy string-brane collisions, JHEP 11 (2013) 126 [arXiv:1310.1254] [INSPIRE].
P. Di Vecchia, A. Luna, S.G. Naculich, R. Russo, G. Veneziano and C.D. White, A tale of two exponentiations in \( \mathcal{N} \) = 8 supergravity, Phys. Lett. B 798 (2019) 134927 [arXiv:1908.05603] [INSPIRE].
P. Di Vecchia, S.G. Naculich, R. Russo, G. Veneziano and C.D. White, A tale of two exponentiations in \( \mathcal{N} \) = 8 supergravity at subleading level, JHEP 03 (2020) 173 [arXiv:1911.11716] [INSPIRE].
Z. Bern, H. Ita, J. Parra-Martínez and M.S. Ruf, Universality in the classical limit of massless gravitational scattering, Phys. Rev. Lett. 125 (2020) 031601 [arXiv:2002.02459] [INSPIRE].
M. Ciafaloni and D. Colferai, Rescattering corrections and self-consistent metric in Planckian scattering, JHEP 10 (2014) 085 [arXiv:1406.6540] [INSPIRE].
B.C. Eu, On the WKB Approximation in Time-Dependent Scattering Theory Including Rearrangement Processes, J. Chem. Phys. 57 (1972) 2531.
S. Caron-Huot and Z. Zahraee, Integrability of Black Hole Orbits in Maximal Supergravity, JHEP 07 (2019) 179 [arXiv:1810.04694] [INSPIRE].
R.N. Lee, LiteRed 1.4: a powerful tool for reduction of multiloop integrals, J. Phys. Conf. Ser. 523 (2014) 012059 [arXiv:1310.1145] [INSPIRE].
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: 2107.12891
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
Damgaard, P.H., Planté, L. & Vanhove, P. On an exponential representation of the gravitational S-matrix. J. High Energ. Phys. 2021, 213 (2021). https://doi.org/10.1007/JHEP11(2021)213
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2021)213