Abstract
We perform a Feynman diagram calculation of the two-loop scattering amplitude for gravitationally interacting massive particles in the classical limit. Conveniently, we are able to sidestep the most taxing diagrams by exploiting the test-particle limit in which the system is fully characterized by a particle propagating in a Schwarzschild spacetime. We assume a general choice of graviton field basis and gauge fixing that contains as a subset the well-known deDonder gauge and its various cousins. As a highly nontrivial consistency check, all gauge parameters evaporate from the final answer. Moreover, our result exactly matches that of Bern et al. [39], here verified up to sixth post-Newtonian order while also reproducing the same unique velocity resummation at third post-Minkowksian order.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
LIGO Scientific, Virgo collaboration, Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett. 116 (2016) 061102 [arXiv:1602.03837] [INSPIRE].
LIGO Scientific, Virgo collaboration, GW170817: observation of gravitational waves from a binary neutron star inspiral, Phys. Rev. Lett. 119 (2017) 161101 [arXiv:1710.05832] [INSPIRE].
J.F. Donoghue, Leading quantum correction to the Newtonian potential, Phys. Rev. Lett. 72 (1994) 2996 [gr-qc/9310024] [INSPIRE].
J.F. Donoghue, General relativity as an effective field theory: the leading quantum corrections, Phys. Rev. D 50 (1994) 3874 [gr-qc/9405057] [INSPIRE].
N.E. Bjerrum-Bohr, J.F. Donoghue and B.R. Holstein, Quantum gravitational corrections to the nonrelativistic scattering potential of two masses, Phys. Rev. D 67 (2003) 084033 [Erratum ibid. 71 (2005) 069903] [hep-th/0211072] [INSPIRE].
B.R. Holstein and A. Ross, Spin effects in long range gravitational scattering, arXiv:0802.0716 [INSPIRE].
N.E.J. Bjerrum-Bohr, J.F. Donoghue and P. Vanhove, On-shell techniques and universal results in quantum gravity, JHEP 02 (2014) 111 [arXiv:1309.0804] [INSPIRE].
Y. Iwasaki, Quantum theory of gravitation vs. classical theory. — Fourth-order potential, Prog. Theor. Phys. 46 (1971) 1587 [INSPIRE].
Y. Iwasaki, Fourth-order gravitational potential based on quantum field theory, Lett. Nuovo Cim. 1S2 (1971) 783.
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 (2017) 104009 [arXiv:1612.00482] [INSPIRE].
H. Okamura, T. Ohta, T. Kimura and K. Hiida, Perturbation calculation of gravitational potentials, Prog. Theor. Phys. 50 (1973) 2066 [INSPIRE].
D. Neill and I.Z. Rothstein, Classical space-times from the S matrix, Nucl. Phys. B 877 (2013) 177 [arXiv:1304.7263] [INSPIRE].
V. Vaidya, Gravitational spin Hamiltonians from the S matrix, Phys. Rev. D 91 (2015) 024017 [arXiv:1410.5348] [INSPIRE].
T. Damour, High-energy gravitational scattering and the general relativistic two-body problem, Phys. Rev. D 97 (2018) 044038 [arXiv:1710.10599] [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].
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].
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, arXiv:1910.09366 [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].
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].
F. Cachazo and A. Guevara, Leading singularities and classical gravitational scattering, JHEP 02 (2020) 181 [arXiv:1705.10262] [INSPIRE].
N.E.J. Bjerrum-Bohr et al., General relativity from scattering amplitudes, Phys. Rev. Lett. 121 (2018) 171601 [arXiv:1806.04920] [INSPIRE].
D.A. Kosower, B. Maybee and D. O’Connell, Amplitudes, observables and classical scattering, JHEP 02 (2019) 137 [arXiv:1811.10950] [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].
A. Guevara, Holomorphic classical limit for spin effects in gravitational and electromagnetic scattering, JHEP 04 (2019) 033 [arXiv:1706.02314] [INSPIRE].
N. Arkani-Hamed, T.-C. Huang and Y.-t. Huang, Scattering amplitudes for all masses and spins, arXiv:1709.04891 [I NSPIRE].
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].
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].
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].
Y.F. Bautista and A. Guevara, From scattering amplitudes to classical physics: universality, double copy and soft theorems, arXiv:1903.12419 [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. 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].
N. Arkani-Hamed, Y.-t. Huang and D. O’Connell, Kerr black holes as elementary particles, JHEP 01 (2020) 046 [arXiv:1906.10100] [INSPIRE].
M.-Z. Chung, Y.-T. Huang and J.-W. Kim, Classical potential for general spinning bodies, arXiv:1908.08463 [INSPIRE].
M.-Z. Chung, Y.-T. Huang and J.-W. Kim, Kerr-Newman stress-tensor from minimal coupling to all orders in spin, arXiv:1911.12775 [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].
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].
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].
F. Pretorius, Evolution of binary black hole spacetimes, Phys. Rev. Lett. 95 (2005) 121101 [gr-qc/0507014] [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 et al., Gravitational wave extraction from an inspiraling configuration of merging black holes, Phys. Rev. Lett. 96 (2006) 111102 [gr-qc/0511103] [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].
J. Droste, The field of n moving centres in Einstein’s theory of gravitation, Proc. Acad. Sci. Amst. 19 (1916) 447.
A. Einstein, L. Infeld and B. Hoffmann, The gravitational equations and the problem of motion, Ann. Math. 39 (1938) 65.
T. Ohta, H. Okamura, T. Kimura and K. Hiida, Physically acceptable solution of Einstein’s equation for many-body system, Prog. Theor. Phys. 50 (1973) 492 [INSPIRE].
P. Jaranowski and G. Schaefer, Third postNewtonian higher order ADM Hamilton dynamics for two-body point mass systems, Phys. Rev. D 57 (1998) 7274 [Erratum ibid. 63 (2001) 029902] [gr-qc/9712075] [INSPIRE].
T. Damour, P. Jaranowski and G. Schaefer, Dynamical invariants for general relativistic two-body systems at the third postNewtonian approximation, Phys. Rev. D 62 (2000) 044024 [gr-qc/9912092] [INSPIRE].
L. Blanchet and G. Faye, Equations of motion of point particle binaries at the third postNewtonian order, Phys. Lett. A 271 (2000) 58 [gr-qc/0004009] [INSPIRE].
T. Damour, P. Jaranowski and G. Schaefer, Dimensional regularization of the gravitational interaction of point masses, Phys. Lett. B 513 (2001) 147 [gr-qc/0105038] [INSPIRE].
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 (2014) 064058 [arXiv:1401.4548] [INSPIRE].
P. Jaranowski and G. Schäfer, Derivation of local-in-time fourth post-Newtonian ADM Hamiltonian for spinless compact binaries, Phys. Rev. D 92 (2015) 124043 [arXiv:1508.01016] [INSPIRE].
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 (2016) 084037 [arXiv:1512.02876] [INSPIRE].
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 (2017) 044026 [arXiv:1610.07934] [INSPIRE].
D. Bini and T. Damour, Gravitational scattering of two black holes at the fourth post-Newtonian approximation, Phys. Rev. D 96 (2017) 064021 [arXiv:1706.06877] [INSPIRE].
L. Bernard et al., Dimensional regularization of the IR divergences in the Fokker action of point-particle binaries at the fourth post-Newtonian order, Phys. Rev. D 96 (2017) 104043 [arXiv:1706.08480] [INSPIRE].
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 (2018) 044023 [arXiv:1707.09289] [INSPIRE].
L. Bernard, L. Blanchet, G. Faye and T. Marchand, Center-of-mass equations of motion and conserved integrals of compact binary systems at the fourth post-Newtonian order, Phys. Rev. D 97 (2018) 044037 [arXiv:1711.00283] [INSPIRE].
B. Bertotti, On gravitational motion, Nuovo Cim. 4 (1956) 898.
R.P. Kerr, The Lorentz-covariant approximation method in general relativity, Nuovo Cim. 13 (1959) 469.
B. Bertotti and J.F. Plebański, Theory of gravitational perturbations in the fast motion approximation, Ann. Phys. 11 (1960) 169.
M. Portilla, Momentum and angular momentum of two gravitating particles, J. Phys. A 12 (1979) 1075 [INSPIRE].
K. Westpfahl and M. Goller, Gravitational scattering of two relativistic particles in postlinear approximation, Lett. Nuovo Cim. 26 (1979) 573.
M. Portilla, Scattering of two gravitating particles: classical approach, J. Phys. A 13 (1980) 3677 [INSPIRE].
L. Bel et al., Poincaré-invariant gravitational field and equations of motion of two pointlike objects: The postlinear approximation of general relativity, Gen. Rel. Grav. 13 (1981) 963 [INSPIRE].
K. Westpfahl, High-speed scattering of charged and uncharged particles in general relativity, Fortschr. Phys. 33 (1985) 417.
T. Ledvinka, G. Schaefer and J. Bicak, Relativistic Closed-Form Hamiltonian for Many-Body Gravitating Systems in the Post-Minkowskian Approximation, Phys. Rev. Lett. 100 (2008) 251101 [arXiv:0807.0214] [INSPIRE].
T. Damour, Gravitational scattering, post-Minkowskian approximation and Effective One-Body theory, Phys. Rev. D 94 (2016) 104015 [arXiv:1609.00354] [INSPIRE].
W.D. Goldberger and I.Z. Rothstein, An effective field theory of gravity for extended objects, Phys. Rev. D 73 (2006) 104029 [hep-th/0409156] [INSPIRE].
J.B. Gilmore and A. Ross, Effective field theory calculation of second post-Newtonian binary dynamics, Phys. Rev. D 78 (2008) 124021 [arXiv:0810.1328] [INSPIRE].
S. Foffa and R. Sturani, Effective field theory calculation of conservative binary dynamics at third post-Newtonian order, Phys. Rev. D 84 (2011) 044031 [arXiv:1104.1122] [INSPIRE].
R.A. Porto and I.Z. Rothstein, Apparent ambiguities in the post-Newtonian expansion for binary systems, Phys. Rev. D 96 (2017) 024062 [arXiv:1703.06433] [INSPIRE].
S. Foffa et al., Static two-body potential at fifth post-Newtonian order, Phys. Rev. Lett. 122 (2019) 241605 [arXiv:1902.10571] [INSPIRE].
J. Blümlein, A. Maier and P. Marquard, Five-loop static contribution to the gravitational interaction potential of two point masses, Phys. Lett. B 800 (2020) 135100 [arXiv:1902.11180].
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 (2020) 115041 [arXiv:2003.01692] [INSPIRE].
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 (2019) 024047 [arXiv:1903.05113] [INSPIRE].
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: Renormalized Lagrangian, Phys. Rev. D 100 (2019) 024048 [arXiv:1903.05118] [INSPIRE].
H. Kawai, D.C. Lewellen and S.H.H. Tye, A relation between tree amplitudes of closed and open strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].
Z. Bern, L.J. Dixon, M. Perelstein and J.S. Rozowsky, Multileg one loop gravity amplitudes from gauge theory, Nucl. Phys. B 546 (1999) 423 [hep-th/9811140] [INSPIRE].
Z. Bern, J.J.M. Carrasco and H. Johansson, New relations for gauge-theory amplitudes, Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE].
Z. Bern, J.J.M. Carrasco and H. Johansson, Perturbative quantum gravity as a double copy of gauge theory, Phys. Rev. Lett. 105 (2010) 061602 [arXiv:1004.0476] [INSPIRE].
Z. Bern et al., The duality between color and kinematics and its applications, arXiv:1909.01358 [INSPIRE].
Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys. B 425 (1994) 217 [hep-ph/9403226] [INSPIRE].
Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys. B 435 (1995) 59 [hep-ph/9409265] [INSPIRE].
Z. Bern, L.J. Dixon and D.A. Kosower, One loop amplitudes for e+ e− to four partons, Nucl. Phys. B 513 (1998) 3 [hep-ph/9708239] [INSPIRE].
R. Britto, F. Cachazo and B. Feng, Generalized unitarity and one-loop amplitudes in N = 4 super-Yang-Mills, Nucl. Phys. B 725 (2005) 275 [hep-th/0412103] [INSPIRE].
Z. Bern, J.J.M. Carrasco, H. Johansson and D.A. Kosower, Maximally supersymmetric planar Yang-Mills amplitudes at five loops, Phys. Rev. D 76 (2007) 125020 [arXiv:0705.1864] [INSPIRE].
A. Antonelli et al., Energetics of two-body Hamiltonians in post-Minkowskian gravity, Phys. Rev. D 99 (2019) 104004 [arXiv:1901.07102] [INSPIRE].
D. Bini, T. Damour and A. Geralico, Novel approach to binary dynamics: application to the fifth post-Newtonian level, Phys. Rev. Lett. 123 (2019) 231104 [arXiv:1909.02375] [INSPIRE].
J.M. Henn and B. Mistlberger, Four-graviton scattering to three loops in \( \mathcal{N} \) = 8 supergravity, JHEP 05 (2019) 023 [arXiv:1902.07221] [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-Martinez and M.S. Ruf, Universality in the classical limit of massless gravitational scattering, arXiv:2002.02459 [INSPIRE].
S. Abreu et al., The two-loop four-graviton scattering amplitudes, Phys. Rev. Lett. 124 (2020) 211601 [arXiv:2002.12374] [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].
T. Damour, Classical and quantum scattering in post-Minkowskian gravity, arXiv:1912.02139 [INSPIRE].
C. Cheung and G.N. Remmen, Hidden simplicity of the gravity action, JHEP 09 (2017) 002 [arXiv:1705.00626] [INSPIRE].
F.A. Berends and W.T. Giele, Recursive calculations for processes with n gluons, Nucl. Phys. B 306 (1988) 759 [INSPIRE].
C. Cheung and G.N. Remmen, Twofold symmetries of the pure gravity action, JHEP 01 (2017) 104 [arXiv:1612.03927] [INSPIRE].
Wolfram Research Inc., Mathematica. Version 12.0, Champaign, U.S.A. (2019).
R. Mertig, M. Böhm and A. Denner, FEYN CALC: computer algebraic calculation of Feynman amplitudes, Comput. Phys. Commun. 64 (1991) 345 [INSPIRE].
V. Shtabovenko, R. Mertig and F. Orellana, New developments in FeynCalc 9.0, Comput. Phys. Commun. 207 (2016) 432 [arXiv:1601.01167] [INSPIRE].
V. Shtabovenko, R. Mertig and F. Orellana, FeynCalc 9.3: new features and improvements, arXiv:2001.04407 [INSPIRE].
Jose M. Martin-Garcia, et al., xAct: efficient tensor computer algebra for the Wolfram language, http://www.xact.es/.
J. Blümlein, A. Maier, P. Marquard and G. Schäfer, Testing binary dynamics in gravity at the sixth post-Newtonian level, Phys. Lett. B 807 (2020) 135496 [arXiv:2003.07145] [INSPIRE].
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
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: 2003.08351
Electronic supplementary material
ESM 1
(GZ 43 kb)
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Cheung, C., Solon, M.P. Classical gravitational scattering at \( \mathcal{O} \)(G3) from Feynman diagrams. J. High Energ. Phys. 2020, 144 (2020). https://doi.org/10.1007/JHEP06(2020)144
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2020)144