Abstract
Amplitude methods have proven to be a promising technique to perform Post-Minkowskian calculations used as inputs to construct gravitational waveforms. In this paper, we show how these methods can be extended beyond the standard calculations in General Relativity with a minimal coupling to matter. As proof of principle, we consider spinless particles conformally coupled to a gravitational helicity-0 mode. We clarify the subtleties in the matching procedure that lead to the potential for conformally coupled matter. We show that in the probe particle limit, we can reproduce well known results for the field profile. With the scattering amplitudes at hand, we compute the conservative potential and scattering angle for the binary system. We find that the result is a non trivial expansion that involves not only the coupling strengths, but also a non trivial dependence on the energy/momentum of the scattered particles.
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].
A. Buonanno and T. Damour, Effective one-body approach to general relativistic two-body dynamics, Phys. Rev. D 59 (1999) 084006 [gr-qc/9811091] [INSPIRE].
A. Buonanno and T. Damour, Transition from inspiral to plunge in binary black hole coalescences, Phys. Rev. D 62 (2000) 064015 [gr-qc/0001013] [INSPIRE].
M. Campanelli, C.O. Lousto, P. Marronetti and Y. Zlochower, Accurate evolutions of orbiting black-hole binaries without excision, Phys. Rev. Lett. 96 (2006) 111101 [gr-qc/0511048] [INSPIRE].
J.G. Baker, J. Centrella, D.-I. Choi, M. Koppitz and J. van Meter, Gravitational wave extraction from an inspiraling configuration of merging black holes, Phys. Rev. Lett. 96 (2006) 111102 [gr-qc/0511103] [INSPIRE].
F. Pretorius, Evolution of binary black hole spacetimes, Phys. Rev. Lett. 95 (2005) 121101 [gr-qc/0507014] [INSPIRE].
Y. Mino, M. Sasaki and T. Tanaka, Gravitational radiation reaction to a particle motion, Phys. Rev. D 55 (1997) 3457 [gr-qc/9606018] [INSPIRE].
T.C. Quinn and R.M. Wald, An Axiomatic approach to electromagnetic and gravitational radiation reaction of particles in curved space-time, Phys. Rev. D 56 (1997) 3381 [gr-qc/9610053] [INSPIRE].
L. Blanchet, Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries, Living Rev. Rel. 17 (2014) 2 [arXiv:1310.1528] [INSPIRE].
G. Schäfer and P. Jaranowski, Hamiltonian formulation of general relativity and post-Newtonian dynamics of compact binaries, Living Rev. Rel. 21 (2018) 7 [arXiv:1805.07240] [INSPIRE].
L. Barack and A. Pound, Self-force and radiation reaction in general relativity, Rept. Prog. Phys. 82 (2019) 016904 [arXiv:1805.10385] [INSPIRE].
L. Barack et al., Black holes, gravitational waves and fundamental physics: a roadmap, Class. Quant. Grav. 36 (2019) 143001 [arXiv:1806.05195] [INSPIRE].
R.A. Porto, The effective field theorist’s approach to gravitational dynamics, Phys. Rept. 633 (2016) 1 [arXiv:1601.04914] [INSPIRE].
M. Levi, Effective Field Theories of Post-Newtonian Gravity: A comprehensive review, Rept. Prog. Phys. 83 (2020) 075901 [arXiv:1807.01699] [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].
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].
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].
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].
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].
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].
F. Cachazo and A. Guevara, Leading Singularities and Classical Gravitational Scattering, JHEP 02 (2020) 181 [arXiv:1705.10262] [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].
P.H. Damgaard, K. Haddad and A. Helset, Heavy Black Hole Effective Theory, JHEP 11 (2019) 070 [arXiv:1908.10308] [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].
D.A. Kosower, B. Maybee and D. O’Connell, Amplitudes, Observables, and Classical Scattering, JHEP 02 (2019) 137 [arXiv:1811.10950] [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].
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].
S. Mougiakakos and P. Vanhove, Schwarzschild-Tangherlini metric from scattering amplitudes in various dimensions, Phys. Rev. D 103 (2021) 026001 [arXiv:2010.08882] [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, A. Luna, R. Roiban, C.-H. Shen and M. Zeng, Spinning black hole binary dynamics, scattering amplitudes, and effective field theory, Phys. Rev. D 104 (2021) 065014 [arXiv:2005.03071] [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].
E. Herrmann, J. Parra-Martinez, M.S. Ruf and M. Zeng, Radiative Classical Gravitational Observables at \( \mathcal{O} \)(G3) from Scattering Amplitudes, arXiv:2104.03957 [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].
G. Kälin and R.A. Porto, From Boundary Data to Bound States, JHEP 01 (2020) 072 [arXiv:1910.03008] [INSPIRE].
G. Kälin and R.A. Porto, From boundary data to bound states. Part II. Scattering angle to dynamical invariants (with twist), JHEP 02 (2020) 120 [arXiv:1911.09130] [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].
C. Cheung and M.P. Solon, Classical gravitational scattering at \( \mathcal{O} \)(G3) from Feynman diagrams, JHEP 06 (2020) 144 [arXiv:2003.08351] [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].
G. Kälin, Z. Liu and R.A. Porto, Conservative Tidal Effects in Compact Binary Systems to Next-to-Leading Post-Minkowskian Order, Phys. Rev. D 102 (2020) 124025 [arXiv:2008.06047] [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].
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].
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].
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 (2021) 024037 [arXiv:2103.14612] [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].
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, arXiv:2106.08276 [INSPIRE].
A. Cristofoli, R. Gonzo, D.A. Kosower and D. O’Connell, Waveforms from Amplitudes, arXiv:2107.10193 [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].
D. Kosmopoulos and A. Luna, Quadratic-in-spin Hamiltonian at \( \mathcal{O} \)(G2) from scattering amplitudes, JHEP 07 (2021) 037 [arXiv:2102.10137] [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, arXiv:2108.02178 [INSPIRE].
A. Brandhuber and G. Travaglini, On higher-derivative effects on the gravitational potential and particle bending, JHEP 01 (2020) 010 [arXiv:1905.05657] [INSPIRE].
M. Accettulli Huber, A. Brandhuber, S. De Angelis and G. Travaglini, Note on the absence of R2 corrections to Newton’s potential, Phys. Rev. D 101 (2020) 046011 [arXiv:1911.10108] [INSPIRE].
W.T. Emond and N. Moynihan, Scattering Amplitudes, Black Holes and Leading Singularities in Cubic Theories of Gravity, JHEP 12 (2019) 019 [arXiv:1905.08213] [INSPIRE].
M. Accettulli Huber, A. Brandhuber, S. De Angelis and G. Travaglini, Eikonal phase matrix, deflection angle and time delay in effective field theories of gravity, Phys. Rev. D 102 (2020) 046014 [arXiv:2006.02375] [INSPIRE].
M. Accettulli Huber, A. Brandhuber, S. De Angelis and G. Travaglini, From amplitudes to gravitational radiation with cubic interactions and tidal effects, Phys. Rev. D 103 (2021) 045015 [arXiv:2012.06548] [INSPIRE].
C. Deffayet, S. Deser and G. Esposito-Farese, Arbitrary p-form Galileons, Phys. Rev. D 82 (2010) 061501 [arXiv:1007.5278] [INSPIRE].
L. Heisenberg, Generalization of the Proca Action, JCAP 05 (2014) 015 [arXiv:1402.7026] [INSPIRE].
G. Tasinato, Cosmic Acceleration from Abelian Symmetry Breaking, JHEP 04 (2014) 067 [arXiv:1402.6450] [INSPIRE].
E. Allys, P. Peter and Y. Rodriguez, Generalized Proca action for an Abelian vector field, JCAP 02 (2016) 004 [arXiv:1511.03101] [INSPIRE].
M. Hull, K. Koyama and G. Tasinato, Covariantized vector Galileons, Phys. Rev. D 93 (2016) 064012 [arXiv:1510.07029] [INSPIRE].
E. Allys, J.P. Beltran Almeida, P. Peter and Y. Rodríguez, On the 4D generalized Proca action for an Abelian vector field, JCAP 09 (2016) 026 [arXiv:1605.08355] [INSPIRE].
L. Heisenberg, R. Kase and S. Tsujikawa, Beyond generalized Proca theories, Phys. Lett. B 760 (2016) 617 [arXiv:1605.05565] [INSPIRE].
J. Beltran Jimenez and L. Heisenberg, Derivative self-interactions for a massive vector field, Phys. Lett. B 757 (2016) 405 [arXiv:1602.03410] [INSPIRE].
A. De Felice, L. Heisenberg, R. Kase, S. Mukohyama, S. Tsujikawa and Y.-l. Zhang, Cosmology in generalized Proca theories, JCAP 06 (2016) 048 [arXiv:1603.05806] [INSPIRE].
E. Allys, Au-delà des modèles standards en cosmologie, Ph.D. Thesis, UPMC, Paris France (2017) [arXiv:1710.02143] [INSPIRE].
C. de Rham and V. Pozsgay, New class of Proca interactions, Phys. Rev. D 102 (2020) 083508 [arXiv:2003.13773] [INSPIRE].
C.M. Will, The Confrontation between General Relativity and Experiment, Living Rev. Rel. 17 (2014) 4 [arXiv:1403.7377] [INSPIRE].
B. Bertotti, L. Iess and P. Tortora, A test of general relativity using radio links with the Cassini spacecraft, Nature 425 (2003) 374 [INSPIRE].
A.I. Vainshtein, To the problem of nonvanishing gravitation mass, Phys. Lett. B 39 (1972) 393 [INSPIRE].
E. Babichev and C. Deffayet, An introduction to the Vainshtein mechanism, Class. Quant. Grav. 30 (2013) 184001 [arXiv:1304.7240] [INSPIRE].
G.R. Dvali, G. Gabadadze and M. Porrati, 4D gravity on a brane in 5D Minkowski space, Phys. Lett. B 485 (2000) 208 [hep-th/0005016] [INSPIRE].
C. de Rham et al., Cascading gravity: Extending the Dvali-Gabadadze-Porrati model to higher dimension, Phys. Rev. Lett. 100 (2008) 251603 [arXiv:0711.2072] [INSPIRE].
C. de Rham, S. Hofmann, J. Khoury and A.J. Tolley, Cascading Gravity and Degravitation, JCAP 02 (2008) 011 [arXiv:0712.2821] [INSPIRE].
C. de Rham, J. Khoury and A.J. Tolley, Flat 3-Brane with Tension in Cascading Gravity, Phys. Rev. Lett. 103 (2009) 161601 [arXiv:0907.0473] [INSPIRE].
C. de Rham and G. Gabadadze, Generalization of the Fierz-Pauli Action, Phys. Rev. D 82 (2010) 044020 [arXiv:1007.0443] [INSPIRE].
C. de Rham, G. Gabadadze and A.J. Tolley, Resummation of Massive Gravity, Phys. Rev. Lett. 106 (2011) 231101 [arXiv:1011.1232] [INSPIRE].
C. Deffayet, G.R. Dvali, G. Gabadadze and A.I. Vainshtein, Nonperturbative continuity in graviton mass versus perturbative discontinuity, Phys. Rev. D 65 (2002) 044026 [hep-th/0106001] [INSPIRE].
C. de Rham, Massive Gravity, Living Rev. Rel. 17 (2014) 7 [arXiv:1401.4173] [INSPIRE].
C. de Rham and R.H. Ribeiro, Riding on irrelevant operators, JCAP 11 (2014) 016 [arXiv:1405.5213] [INSPIRE].
A. Joyce, B. Jain, J. Khoury and M. Trodden, Beyond the Cosmological Standard Model, Phys. Rept. 568 (2015) 1 [arXiv:1407.0059] [INSPIRE].
C. de Rham, A.J. Tolley and D.H. Wesley, Vainshtein Mechanism in Binary Pulsars, Phys. Rev. D 87 (2013) 044025 [arXiv:1208.0580] [INSPIRE].
Y.-Z. Chu and M. Trodden, Retarded Green’s function of a Vainshtein system and Galileon waves, Phys. Rev. D 87 (2013) 024011 [arXiv:1210.6651] [INSPIRE].
C. de Rham, A. Matas and A.J. Tolley, Galileon Radiation from Binary Systems, Phys. Rev. D 87 (2013) 064024 [arXiv:1212.5212] [INSPIRE].
F. Dar, C. De Rham, J.T. Deskins, J.T. Giblin and A.J. Tolley, Scalar Gravitational Radiation from Binaries: Vainshtein Mechanism in Time-dependent Systems, Class. Quant. Grav. 36 (2019) 025008 [arXiv:1808.02165] [INSPIRE].
P. Brax, A.-C. Davis and R. Jha, Neutron Stars in Screened Modified Gravity: Chameleon vs. Dilaton, Phys. Rev. D 95 (2017) 083514 [arXiv:1702.02983] [INSPIRE].
A. Kuntz, Two-body potential of Vainshtein screened theories, Phys. Rev. D 100 (2019) 024024 [arXiv:1905.07340] [INSPIRE].
B.F. de Aguiar and R.F.P. Mendes, Highly compact neutron stars and screening mechanisms: Equilibrium and stability, Phys. Rev. D 102 (2020) 024064 [arXiv:2006.10080] [INSPIRE].
M. Bezares, L. ter Haar, M. Crisostomi, E. Barausse and C. Palenzuela, Kinetic screening in nonlinear stellar oscillations and gravitational collapse, Phys. Rev. D 104 (2021) 044022 [arXiv:2105.13992] [INSPIRE].
P. Brax, L. Heisenberg and A. Kuntz, Unveiling the Galileon in a three-body system: scalar and gravitational wave production, JCAP 05 (2020) 012 [arXiv:2002.12590] [INSPIRE].
C. Renevey, R. McManus, C. Dalang and L. Lombriser, The effect of screening mechanisms on black hole binary inspiral waveforms, arXiv:2106.05678 [INSPIRE].
A. Dima, M. Bezares and E. Barausse, Dynamical chameleon neutron stars: Stability, radial oscillations, and scalar radiation in spherical symmetry, Phys. Rev. D 104 (2021) 084017 [arXiv:2107.04359] [INSPIRE].
M. Bezares, R. Aguilera-Miret, L. ter Haar, M. Crisostomi, C. Palenzuela and E. Barausse, No evidence of kinetic screening in merging binary neutron stars, arXiv:2107.05648 [INSPIRE].
A. Nicolis, R. Rattazzi and E. Trincherini, The Galileon as a local modification of gravity, Phys. Rev. D 79 (2009) 064036 [arXiv:0811.2197] [INSPIRE].
M.A. Luty, M. Porrati and R. Rattazzi, Strong interactions and stability in the DGP model, JHEP 09 (2003) 029 [hep-th/0303116] [INSPIRE].
A. Nicolis and R. Rattazzi, Classical and quantum consistency of the DGP model, JHEP 06 (2004) 059 [hep-th/0404159] [INSPIRE].
C. de Rham, Massive gravity from Dirichlet boundary conditions, Phys. Lett. B 688 (2010) 137 [arXiv:0910.5474] [INSPIRE].
C. de Rham and G. Gabadadze, Selftuned Massive Spin-2, Phys. Lett. B 693 (2010) 334 [arXiv:1006.4367] [INSPIRE].
C. de Rham and A.J. Tolley, DBI and the Galileon reunited, JCAP 05 (2010) 015 [arXiv:1003.5917] [INSPIRE].
M. Beneke and V.A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B 522 (1998) 321 [hep-ph/9711391] [INSPIRE].
V.A. Smirnov, Evaluating Feynman integrals, in Springer Tracts in Modern Physics 211, Springer (2005) [INSPIRE].
G. Passarino and M.J.G. Veltman, One Loop Corrections for e+e− Annihilation Into μ+μ− in the Weinberg Model, Nucl. Phys. B 160 (1979) 151 [INSPIRE].
Y. Iwasaki, Quantum theory of gravitation vs. classical theory: fourth-order potential, Prog. Theor. Phys. 46 (1971) 1587 [INSPIRE].
D. Neill and I.Z. Rothstein, Classical Space-Times from the S Matrix, Nucl. Phys. B 877 (2013) 177 [arXiv:1304.7263] [INSPIRE].
T. Damour, High-energy gravitational scattering and the general relativistic two-body problem, Phys. Rev. D 97 (2018) 044038 [arXiv:1710.10599] [INSPIRE].
A. Paliov and S. Rosendorff, High-energy phase shifts produced by repulsive singular potentials, J. Math. Phys. 8 (1967) 1829.
S.J. Wallace, Eikonal expansion, Annals Phys. 78 (1973) 190 [INSPIRE].
D. Bohm, Quantum Theory, Dover Publications (1989).
T.W. Murphy Jr. et al., APOLLO: millimeter lunar laser ranging, Class. Quant. Grav. 29 (2012) 184005 [INSPIRE].
G. Dvali, A. Gruzinov and M. Zaldarriaga, The Accelerated universe and the moon, Phys. Rev. D 68 (2003) 024012 [hep-ph/0212069] [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: 2107.11384
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
González, M.C., de Rham, C. & Tolley, A.J. Scattering amplitudes for binary systems beyond GR. J. High Energ. Phys. 2021, 87 (2021). https://doi.org/10.1007/JHEP11(2021)087
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2021)087